1996, vol. 11, no. 4, 253{282 de ... - bruno de finetti finetti... · 254 d. m. cifarelli and e....

Statistical Science1996, Vol. 11, No. 4, 253–282

De Finetti’s Contribution to Probabilityand StatisticsDonato Michele Cifarelli and Eugenio Regazzini

Abstract. This paper summarizes the scientific activity of de Finetti inprobability and statistics. It falls into three sections: Section 1 includesan essential biography of de Finetti and a survey of the basic features ofthe scientific milieu in which he took the first steps of his scientific ca-reer; Section 2 concerns de Finetti’s work in probability: (a) foundations,(b) processes with independent increments, (c) sequences of exchangeablerandom variables, and (d) contributions which fall within other fields;Section 3 deals with de Finetti’s contributions to statistics: (a) descrip-tion of frequency distributions, (b) induction and statistics, (c) probabilityand induction, and (d) objectivistic schools and theory of decision. Manyrecent developments of de Finetti’s work are mentioned here and brieflydescribed.

Key words and phrases: Associative mean, Bayesian nonparametricstatistics, Bayes–Laplace paradigm, completely additive probabilities,correlation and monotone dependence, exchangeable and partially ex-changeable random variables, finitely additive probabilities, gambler’sruin, Glivenko–Cantelli theorem, infinitely decomposable laws, predic-tive inference, prevision, principle of coherence processes with indepen-dent increments, reasoning by induction, statistical decision, subjectiveprobability, utility function

This paper includes the text of two lectures thatwe had the honor to give at the Second Interna-tional Workshop on Bayesian Robustness, on May22, 1995.

0. INTRODUCTION

This is to mark the tenth anniversary of deFinetti’s death on 20 July 1985. We were thus de-prived of his sincere and discreet friendship, of hiscritical and constructive insights, of his passionateand disinterested battle for a fairer social order. Heis still very much with us, though, with his impres-sive scientific corpus scattered in more than 290writings.

Donato Michele Cifarelli and Eugenio Regazzini areProfessors of Statistics, Istituto di Metodi Quan-titativi, Universita “L. Bocconi,” v. Sarfatti 25,20136 Milano. Donato Michele Cifarelli is Headof this Institute and Eugenio Regazzini is Direc-tor, Istituto per le Applicazioni della Matematica edell’Informatica (IAMI-CNR), v. Ampere 56, 20131Milano.

The Scientific Committee of the Second Interna-tional Workshop on Bayesian Robustness thought itfitting to devote a special session to a retrospectiveanalysis of de Finetti’s scientific endeavor. We havethe honor and pleasure of delivering two lectures onde Finetti’s contributions to probability and statis-tics. In point of fact, we have partly enlarged thescope of our original plan by inserting also a surveyof the Italian scientific circle which was closest toprobability and statistics when de Finetti embarkedon his venture into the subjects.

Recently, Barlow has written that the publicationof de Finetti’s paper La prevision (1937a) was thefirst major event that brought about the rebirth ofBayesian statistics, and that:

Despite the growing number of papersand books which have been influencedby de Finetti, his overall influence isstill minimal. Why is this? Perhaps onereason is communication. De Finetti’sstyle of writing is difficult to understand,even for Italian mathematicians and fewEnglish-speaking mathematicians have

253

254 D. M. CIFARELLI AND E. REGAZZINI

really tried to interpret what he haswritten. [See Barlow, 1992, page 131.]

As a matter of fact, the language of Brunode Finetti is elaborate, full of nuances and perfectlyappropriate for the manifestation of his sharp wayof thinking. It is then difficult to translate it intoa language other than the original without losingsome of its peculiar stylistic aspects or introducingsome simplifications which may distort the originalmeaning. Even if this kind of imperfection is some-times present in translations of de Finetti’s works,yet we are grateful to the scholars who have helpedbring de Finetti’s contributions to the attention ofthe international scientific community.

We have taken upon ourselves the task of summa-rizing the scientific activity of de Finetti in proba-bility and statistics, hoping to help spread his majorideas and change the picture portrayed by Barlow.In going through de Finetti’s original papers, wehave found it convenient not to linger on well-knownworks such as Teoria delle Probabilita (1970), butrather on a number of less known earlier paperswhich blazed the trail of de Finetti’s scientificventure.

To get one’s bearings in his copious productionis no easy matter, and we are indebted to previouswork, mentioned in the General References section,for a number of bibliographic hints. In particular,de Finetti’s complete bibliography can be found inDaboni (1987).

Many recent developments of de Finetti’s workare mentioned here and briefly described, as are thelengthier developments, in note within the text.

1. HISTORICAL NOTES

1.1 Essential Biography

De Finetti was born of Italian parents on 13 June1906 in Innsbruck (Austria). At 17 he enrolled at thePolytechnic of Milan with a view to obtaining a de-gree in engineering. During his third year of study(in Italy engineering is a five-year course), a Facultyof Mathematics was opened in Milan. He attendeda few classes there, and, following his more theoret-ical bent, shifted subjects. In 1927 he obtained a de-gree in applied mathematics. He discussed his grad-uation dissertation on affine geometry with GiulioVivanti, a mathematician known for some notewor-thy contributions to complex analysis. Soon after-wards, de Finetti accepted a position in Rome, at theIstituto Centrale di Statistica, presided over, at thattime, by an outstanding Italian statistician: CorradoGini. De Finetti worked there until 1931. In thoseyears, he laid the foundations for his principal con-

tributions to probability theory and statistics: thesubjective approach to probability; definition andanalysis of sequences of exchangeable events; def-inition and analysis of processes with stationary in-dependent increments and infinitely decomposablelaws; theory of mean values. In 1930 de Finetti qual-ified in a competition as a university lecturer ofMathematical Analysis.

In 1931, he moved to Trieste, where he accepteda position as actuary at the Assicurazioni Gener-ali. Trieste was included in the Austro-HungarianEmpire until 1918. De Finetti’s father and grand-father had worked as engineers there. In 1906 hisfather was actually engaged in the construction ofnew railway lines in the Innsbruck area (Upper Ty-rol). During his stay in Trieste, de Finetti developedthe research he had started in Rome, but he alsoobtained significant results in actuarial and finan-cial mathematics as well as in economics. In addi-tion, he was active in the mechanization of someactuarial services. It was probably this operationalbackground which later enabled him to understandthe revolutionary impact of the computer on scien-tific calculus. As a matter of fact, he was one ofthe first mathematicians in Italy able to solve prob-lems of numerical analysis by means of a computer(cf. Fichera, 1987). In spite of his impending actu-arial activity, in Trieste de Finetti also started histeaching career (mathematical analysis, actuarialand financial mathematics, probability), with a fewyears’s stint at the renowned University of Padua.However, it was only in 1947 that he obtained hischair as full Professor of Financial Mathematics inTrieste. He had actually won it in a nationwide com-petition back in 1939, but at that time he was un-married and, by a law in force in those days in Italy,bachelors were not permitted to hold any position inthe public service. Such a law was cancelled by theAllies, who ruled over Trieste from the end of thewar (April 1945) until 1954.

In 1954 de Finetti moved to the Faculty of Eco-nomics at the University of Rome “La Sapienza.” In1961, he changed to the Faculty of Sciences in Rome,where he was Professor of the Theory of Probabilityuntil 1976.

1.2 Probability and Statistics in Italy Duringthe 1920s and 1930s

De Finetti did not attend any course in proba-bility or statistics. With the exception of the Uni-versity of Rome, in the 1920s no Italian faculty ofmathematics included those subjects in its course ofstudies. Around 1925, de Finetti was attracted byprobability, indirectly, through the reading of a pa-per on Mendelian heredity by C. Foa, a biologist. De

DE FINETTI’S CONTRIBUTIONS 255

Finetti started research on the subject, reaching re-sults that Foa, Vivanti and Mortara (a statistician,at that time, active in Milan) considered of interest.Mortara submitted part of de Finetti’s research forpublication in Metron, to Gini (editor and owner ofthat journal), who published the paper in an issueof 1926. Other parts of the same research were pub-lished in the proceedings of the Reale Accademia deiLincei in 1927, with nomination by Foa. These arethe first three items in the long list of de Finetti’sscientific works.

The above-mentioned genetics problem led himto go deeper into probability and statistics. By theway, from some of his written recollections, we knowthat he studied probability in Czuber’s Wahrschein-lichkeitslehre and in Castelnuovo’s Calcolo delleProbabilita e Applicazioni (Castelnuovo, 1919). Theformer is well known to statisticians, while the lat-ter is more renowned among pure mathematicians,following his outstanding contributions to geometry.In point of fact, Castelnuovo played a fundamentalrole in introducing probability into the culture ofofficial Italian mathematical circles.

Thanks to the initiative of Guido Castelnuovo(1865–1952), the Faculty of Mathematics at theUniversity of Rome, starting from 1915, introducedcourses of probability and actuarial mathematics.They represented the germ of the School of Statis-tics and Actuarial Sciences, officially established in1927. Castelnuovo was the first head of that school.Among its students, a few became well known fortheir remarkable contributions to probability the-ory and its applications, for example, Onicescu andMihok. In 1935 that school merged with the SpecialSchool of Statistics imbued with Gini’s line of think-ing in statistics, to found the present-day Facultyof Statistics in Rome. That very same year, Castel-nuovo gave up teaching. In 1938, like all ItalianJews he bore the brunt of Fascist racial laws.

The first edition of Castelnuovo’s book on prob-ability dates back to 1919. Considering the timeit was conceived, this treatise looks quite accuratefrom the mathematical angle, and most up-to-date.In particular, it bears witness to the influence ofthe Russian School of St. Petersburg (Chebyshev,Markov, Lyapunov) on probability theory. Casteln-uovo’s stance as to the foundations of probabilitycan be included under the heading of the empiricalapproach (empirical law of chance), the same posi-tion as Levy’s, Frechet’s and Cantelli’s. Cantelli wasactually Castelnuovo’s most important collaboratorin the drafting of the book. As far as the relation-ship between probability and inductive reasoning isconcerned, Castelnuovo went deeper into the sub-ject. In compliance with his approach to probability,

he rejected the Bayesian formulation of the prob-lem at issue. He proposed to deal with the questionof rejecting an isolated hypothesis on the ground ofits sole likelihood. In this sense, he could be consid-ered as a forerunner of Fisher’s viewpoint on induc-tive reasoning. In any case, Castelnuovo, becauseof his preference for pure mathematics, refrainedfrom taking sides in the controversy on the foun-dations of probability and statistics. He disinterest-edly helped not only scholars who followed the mainlines of his approach, but also those, like de Finetti,who supported alternative viewpoints; see de Finetti(1976).

Francesco Paolo Cantelli (1875–1966) was thefirst “modern” Italian probabilist. His name isdefinitively linked with some fundamental resultsabout the convergence of sequences of random vari-ables: the very famous 0–1 law, strong laws of largenumbers (Cantelli, 1917a, b), the fundamental the-orem of mathematical statistics (Cantelli, 1933b)and a formulation of the law of the iterated loga-rithm (Cantelli, 1933a) which extends Kolmogorov’sversion of the same law, under suitable conditions;see Epifani and Lijoi (1995).

Cantelli’s point of view about the interpretation ofprobability substantially agrees with Castelnuovo’s.In fact, he had no genuine interest in the contro-versy about the foundations of probability. Cantelli’sconception of science was extremely positive and op-erative, in compliance with his position as an ac-tuary at a very high international standard level.Here is how Harald Cramer portrays Cantelli (Weg-man, 1986):

I should mention also another name,the Italian mathematician Cantelli, withwhom we also had contact in our Stock-holm probabilistic group. Cantelli, likemyself, was also working as an actu-ary. As a matter of fact, he had been,among other things, the actuary of thepension board of what was then calledthe Society of Nations in Geneva. Whenhe resigned his position, I became hissuccessor. That gave me a contact withCantelli which I value very much. Youknow his name from the Borel–Cantellicondition. He had written several veryvaluable papers on probability, paperswhich have perhaps not received quitethe attention that they really do de-serve. He was a very temperamentalman. When he was excited, he could cryout his views with his powerful voice: avery energetic fellow.


The extroversion and the determination of Can-telli’s attitudes are confirmed by Gaetano Fichera.They set off the low profile and shy attitudes of deFinetti, so as to conclude that Cantelli was the pho-tographic negative of de Finetti; see Fichera (1987).But, although Cantelli had no inclination for philo-sophical speculation, he stressed the necessity fora formal definition of probability. It could be takenas the starting point for a rigorous construction ofprobability theory considered as a formal theory.On this subject, he formulated an abstract theory ofprobability—shortly before the publication, in 1933,of Kolmogorov’s Grundbegriffe—whose axioms es-sentially coincide with those propounded by thegreat Russian mathematician; see Cantelli (1932)and Ottaviani (1939).

We cannot conclude these notes without quotingCantelli’s initiative leading to the publication of theGiornale dell’Istituto Italiano degli Attuari, in 1930.During his editorship, the Giornale was one of themost prestigious journals dealing with probabil-ity, statistics and actuarial mathematics, and couldcount on the collaboration of outstanding scholars.In particular, a number of de Finetti’s writings ap-peared there, although none of them actually dealwith the foundations of probability. An explana-tion for this can probably be found in Fichera’sabove-quoted paper, when he recollects that speak-ing to Cantelli about subjective probability andfinitely additive probabilities was tantamount topulling a tiger by its tail.

Passing from probability to statistics, the Italianstatistical panorama during the 1920s and 1930swas dominated by Corrado Gini (1884–1965). Asa matter of fact, Gini deeply influenced Italianstatistics until the end of the 1960s. His culturalinterests were deep and broad, so that he handeddown striking applications of statistics to manyaspects of the real world. He was accustomed togearing his statistical methodology toward the spe-cific problem involved under each circumstance.This approach—setting up instruments and tech-niques best befitting specific targets—was alwaysactually oriented by an implicit intuitive overallvision. Indeed, starting from the 1940s, he concen-trated on “a critical work of synthesis drawing itsform and substance from the particular techniquesworked out”; see Castellano (1965). According tothis author, the main works of Gini’s early periodmay be grouped under several headings, namely,the theory of means, the theory of variability andthe theory of statistical relations. In particular, letus recall Gini’s proposal of the mean difference 1 asa general measure of the variability in a frequencydistribution F and of the concentration ratio G (ob-

tained by dividing the mean difference by twice thearithmetic mean) as a measure of the particular as-pect of variability, for nondegenerate distributionswith positive support, known as concentration:

1 =∫

R2|x− y| dF�x�dF�y�

G = 1

2µ;

where

F�x� = 0 for x < 0

and

µ =∫�0;+∞�

xdF�x�

(cf. Stuart and Ord, 1994). Gini introduced 1 in 1912as a measure of the variability, independently of pre-vious works of the astronomers Jordan, von Andraeand Helmert who used the mean difference for quitedifferent reasons.

In a series of papers dating back to the quadri-ennium 1914–1917, Gini shaped a systematicalmethodology for the analysis of relations betweentwo statistical characteristics, which brought clarityand order into a field to whose progress outstandingcontributions had already been made by Bravais,Galton, Pearson and Yule.

Insofar as statistical inference is concerned,Gini deemed Fisher’s approach logically weak andopposed it vigorously. Nonetheless, he refusedthe Neyman–Pearson approach. In the above-mentioned paper, Castellano summarizes Gini’sposition in the following terms:

With penetrating logic Gini proved : : :how important and unavoidable are priorprobabilities in any judgement on themeasures deriving from a sample; he re-stored to induction its essential characterof a conclusion based upon an experience(the facts) and an independent and pre-liminary a priori assumption by whichthe facts can be interpreted.

Gini rendered explicit this position, which ba-sically supports the Bayes–Laplace paradigm, in1939, on the occasion of the first meeting of the So-cieta Italiana di Statistica, in an address titled Ipericoli della statistica. As a matter of fact, in 1911,he had made an application of that paradigm to thestudy of the sex ratio at birth, after determiningthe conditional probability of k2 successes in n1 fu-ture trials given k1 successes in n1 past trials, whenthe prior distribution of the unknown Bernoulli pa-rameter is beta. On the other hand, consistently


with his objective interpretation of probability, Ginideveloped a skeptical attitude toward the value ofstatistical inferences. This, in addition to a pene-trating and proud freedom of thought, preventedhim from being a follower of the neo-Bayesian wayof thinking. He maintains that probability is a fre-quency: not an ordinary frequency, nor a limitingfrequency at that. Probability is the frequency of aphenomenon in the totality of cases in which thatvery phenomenon could occur. In any case, what isinvolved is a finite collective, defined as the classof all cases with respect to which we deem it fit toregulate our conduct.

The above survey should give you an idea of thestudies in probability and statistics in Italy in thedays when de Finetti began tackling those issues,and, consequently, draw your attention to his or-thogonal position with respect to the prevailing wayof thinking. In 1976, with reference to Savage’s co-operation, de Finetti wrote:

I must stress that I owe it to him if mywork is no longer considered as a blas-phemous but harmless heresy, but as aheresy with which the official statisticalchurch is being compelled, unsuccess-fully, to come to terms.

To complete the picture, let us now briefly analyzethe connections between the small coterie of Italianscholars dealing with probability and statistics andforeign schools.

1.3 Connections with Foreign Schools

De Finetti started dealing with probability andstatistics in a period of tremendous developmentsfor the subjects. For instance, Kolmogorov (1903–1987) and Levy (1886–1971) were making their de-cisive contribution to the modern theory of probabil-ity, and Fisher (1890–1962) was setting out the basictechnical concepts for his new approach to statistics.Castelnuovo, Cantelli, Gini and young de Finetti gotinvolved in this impressive cultural revival: on oneside, with their first-hand scientific contributions,and, on the other side, with the opportunity theygave to distinguished foreign scholars of publishingthe new results of their research in Italian scientificjournals.

For example in Metron, the journal founded byGini in 1920, Fisher gave striking examples of hisart of calculating explicit sampling distributions,providing the distribution of F and of its logarithmZ (Fisher, 1921) and an interesting Note on Stu-dent’s distribution (Fisher, 1925). Romanowsky, intwo papers appearing in 1925 and 1928, respec-tively, determined the moments of the standard de-

viation and of the correlation coefficient in normalsamples, and masterfully dealt with a criterion thattwo given samples belong to the same normal dis-tribution.

Castelnuovo presented three important papers ofKolmogorov’s for publication in Atti della Reale Ac-cademia dei Lincei, from 1929 to 1932, dealing withthe strong law of large numbers (Kolmogorov, 1929),the general representation of an associative meanin a discrete distribution (Kolmogorov, 1930) andthe representation of the characteristic functionof an infinitely decomposable law with finite vari-ance (Kolmogorov, 1932a, b). The last two papersare closely connected with de Finetti’s research. Inparticular, they originate from de Finetti’s studieson processes with independent increments and, inturn, represent the starting point of Levy’s funda-mental paper, published in 1934 in Annali dellaScuola Normale Superiore di Pisa. This is thepaper in which Levy—in addition to the general-ization of the Kolmogorov representation—statesthe renowned decomposition of a process withindependent increments.

The publication of the Giornale dell’Istituto Ital-iano degli Attuari (GIIA), starting from 1930 underCantelli’s editorship, made the submission of for-eign works more matter-of-fact. Let us mentionGlivenko’s paper, which appeared in 1933 in GIIA,dealing with the almost-sure uniform convergenceof the empirical distribution function Fn to F, ina sequence of i.i.d. random variables with commoncontinuous distribution function F. In the sameyear, GIIA published Kolmogorov’s paper in whichthe asymptotic distribution of supx

√n�Fn�x�−F�x��

is stated. In 1935, GIIA accepted Neyman’s paperon the factorization theorem. An interesting re-sult reached by Romanowsky, dealing with theasymptotic behaviour of the posterior distribution,is included in an issue of 1931. Other first-ratecontributions in the domain of probability whichappeared in GIIA in the 1930s are noteworthy:Khinchin (1932a), on stationary sequences of ran-dom events; Khinchin (1935), including the charac-terization of the domain of attraction of the normallaw; Khinchin (1936), dealing with a strong lawof large numbers; Levy (1931a), on the stronglaw of large numbers; Levy (1935), on the applica-tion of the geometry of Hilbert spaces to sequencesof random variables; Cramer (1934), focussing on aversion of the strong law of large numbers; Cramer(1935), providing asymptotic expansions of a distri-bution function in series of Hermite polynomials;Glivenko (1936), on a useful refinement of Levy’scontinuity theorem; Slutsky (1934), on some basicresults on stationary (in a wide sense) processes;


and Slutsky (1937), on the foundations of the the-ory of random functions according to a point of viewsimilar to that in Doob (1953). This very same pa-per includes the original version of the celebratedcriterion for the continuity of a random functiondue to Kolmogorov.

Let us conclude with a look at the InternationalCongress of Mathematicians held in Bologna, in1928. Among the statisticians and probabilists whoattended that congress, let us mention B. Hostin-sky, M. Frechet, E. Borel, P. Levy, G. Darmois,C. Gini, C. Bonferroni, G. Castelnuovo, J. Ney-man, A. Lomnicki, S. Bernstein, G. Polya, F. P.Cantelli, B. de Finetti, R. A. Fisher, E. J. Gumbel,C. Jordan, A. Khinchin, O. Morgenstern, E. Slutsky,O. Onicescu, V. Romanowsky and F. M. Urban. TheBologna meeting gave de Finetti a good opportunityto get in touch with the international mathemat-ical community of the time. In a communicationparticularly appreciated by Frechet, Khinchin andNeyman, he summarized his recent research onthe representation of the law of a sequence of ex-changeable events. In addition to the proceedingsof the congress, let us mention a stimulating recentpaper by Seneta (1992) dealing with a controversybetween Cantelli and Slutsky, which occured duringthe Bologna Congress, over priority for the stronglaw of large numbers.

After sketching out the basic features of the sci-entific milieu in which de Finetti took the first stepsof his scientific career in probability and statistics,let us review some of his main contributions.

2. DE FINETTI’S WORK IN PROBABILITY

Three topics stand out in de Finetti’s contribu-tions to probability: the foundations of probability;processes with independent increments; and se-quences of exchangeable variables.

2.1 Foundations of Probability

De Finetti had examined this subject in depthever since his early approach to probability. Infact, in 1931, he was in a position to explain hissubjectivistic point of view in a nearly definitiveway (cf. de Finetti, 1931a, b). Before the appear-ance of these papers, he had concisely stated hisoutlook on probability on the occasion of a politedispute with Maurice Frechet (Frechet, 1930a, b,and de Finetti, 1930b, c), which originated from apaper of de Finetti’s on the limit operation in thetheory of probability; see de Finetti (1930a). In thatpaper, de Finetti, starting from the assumption thatprobability need not be completely additive, devel-

ops some critical remarks on the usual formulationof Cantelli’s theorem, according to which:

Theorem. Let �Xn�n≥1 be a sequence of �0;1�-valued independent and identically distributed ran-dom variables defined on the completely additiveprobability space ��;F ;P�; such that P�X1 = 1� =p. Then

limN→∞

P

(supn≥N

∣∣∣∣1n

n∑k=1

Xk − p∣∣∣∣ ≤ ε

)

= P(

limn→+∞

1n

n∑k=1

Xk = p)

(1)

= 1:(2)

Statements (1) and (2) hold if P is completely ad-ditive but they need not hold if P is not completelyadditive. In any case, one has the following:

Theorem. Given ε, η > 0, there is N0 =N0�ε;η�such that

�3�P

(sup

N≤n≤N+M

∣∣∣∣1n

n∑k=1

Xk − p∣∣∣∣ ≤ ε

)> 1− η;

N ≥N0; M ∈ N:

However, (3), generally speaking, does not entail

P

(supn≥N

∣∣∣∣1n

n∑k=1

Xk − p∣∣∣∣ ≤ ε

)> 1− η; N ≥N0:

Moreover, equality (1) need not occur.De Finetti explains these facts by giving an inter-

esting example, which we report here. Let

�={

12y 1

4;

34y 1

8;

38;

58;

78y : : : y 1

2n;

32n; : : : ;

2n − 12ny : : :

}:

Writing each term of � in its terminating binaryexpansion, we obtain the following representationfor the elements of �:

1 0 0 0 0 · · ·0 1 0 0 0 · · ·1 1 0 0 0 · · ·0 0 1 0 0 · · ·0 1 1 0 0 · · ·1 0 1 0 0 · · ·1 1 1 0 0 · · ·· · · · · · · · · · · · · · · · · ·

Now, for each k in N, define Xk:�→ �0;1� to bethe kth digit of ω in �. Clearly,

�4� limn→+∞

1n

n∑k=1

Xk�ω� = 0 for each ω in �


and

�5�supn≥N

∣∣∣∣1n

n∑k=1

Xk�ω� −12

∣∣∣∣ =12

for each ω in � and N in N:

Moreover, define

�n ={

12;

14;

34; : : : ;

12n; : : : ;

2n − 12n

};

Pn��ω�� =

12n − 1

; if ω ∈ �n;

0; if ω ∈ � \�n;

for each n in N. Then

Pn�Xk = 1� =

0; if k ≥ n+ 1;

2n−1

2n − 1; if k ≤ n;

Pn(�Xk = 1� ∩ �Xi = 1�

)

=

0; if �k ≥ n+ 1� ∨ �i ≥ n+ 1�2n−2

2n − 1; if �k ≤ n� ∧ �i ≤ n�

k 6= i;

:::

Pn

( i⋂j=1

�Xkj= 1�

)=

0; ifi∨

j=1

�kj ≥ n+ 1�;

2n−i

2n − 1; if

i∧j=1

�kj ≤ n�;

k1 6= k2 6= · · · 6= ki

Let us now consider any (finitely additive) proba-bility P on P �� such that P�A� = limn→+∞Pn�A�for every A in Pl�� x= �A ⊂ �: limn→+∞Pn�A�exists�. Then �Xn�n≥1 with respect to such a Pis a sequence of independent and identically dis-tributed random variables with two values (0 and1) and P�X1 = 1� = 1/2. Hence, (3) holds withp = 1/2 (weak formulation of the uniform conver-gence in probability) in spite of (4), which statesthat �∑n

k=1Xk/n�n≥1 converges to 0 in the senseof pointwise convergence. Moreover, (5) shows thatthe usual strong formulation of the Cantelli lawof large numbers does not hold with respect to P.See also Ramakrishnan and Sudderth (1988) for asimilar example.

These facts merely show that a family of consis-tent finite-dimensional probability distributions, ifcomplete additivity is not assumed as a requisiteof probability, does not determine the probability of

“infinitary events” of the type{

limn→+∞

n∑k=1

Xk/n = E�X1�};

{supn≥N

∣∣∣∣1n

n∑k=1

Xk −E�X1�∣∣∣∣ ≤ ε

};

and so on. Those very same facts reveal that thevalidity of strong laws cannot be ascribed to reasonsof physical and objective nature, but to the arbitrarychoice of a coherent extension to “infinitary events”of a probability law assessed on “finitary events”only.

In the same paper, de Finetti dwells upon the fol-lowing circumstance. If additivity of probability isnot complete, then almost-sure convergence of a se-quence of random variables to X does not implyconvergence in law of the same sequence to X (theprevious example, in which X ≡ 0 and the sequenceconverges in law to Y ≡ 1/2). This problem, atthat time, was of particular interest to him, sinceit was related to his research on processes with in-dependent increments. So he stated that, in orderthat almost sure convergence of �Xn�n≥1 to X im-ply convergence in law of the same sequence to X,it suffices that �Xn�n≥1 converges in probability toX. And that is indeed a revolution when you con-sider the run-of-the-mill analysis of stochastic con-vergence! Maybe, in studying consistency of statis-tical procedures, for instance, statisticians ought toponder these “paradoxical” conclusions.

In those months, Frechet had submitted a pa-per for publication in Metron, summarizing thecontent of his 1929 and 1930 courses on stochas-tic convergence, from the σ-additive point of view.This circumstance made the French mathematicianparticularly susceptible to de Finetti’s remarks.Thus, he sent a very short Note to Rendiconti delReale Istituto Lombardo di Scienze e Lettere, inwhich de Finetti’s paper had been published, tosay that de Finetti’s examples were interestingbut, on the other hand, meaningless once the ex-tension of the additivity condition is assumed; seeFrechet (1930a). De Finetti answered Frechet’sobjections (de Finetti, 1930b) by stating that theproblems concerning stochastic convergence aremere signs of a deeper problem concerning the cor-rect mathematical definition of probability. In fact,in his opinion, this definition has to adhere to thenotion of probability as it is conceived by all of usin everyday life, and he preannounces the publica-tion of Probabilismo and Fondamenti per la teoriadelle probabilita, devoted to the philosophic andmathematical aspects, respectively, of the definitionof probability. He maintains that one has anyway


no right to make arbitrary use of the propertiesintroduced to give a mathematical definition ofprobability. Indeed, these very properties have to benot only formally consistent but also intrinsicallynecessary with respect to a meaningful interpre-tation of probability. He goes on to state that thedefinition he intends to make is not necessarilyconducive to a reduction of all admissible probabili-ties to the one family of σ-additive probabilities. Inthis light, his remarks in Sui passaggi al limite (deFinetti, 1930a) do make sense, unless one rejectsthe above definition a priori. Moreover, even if oneshould confine oneself strictly to the formal aspectsof the problem, the restrictive stance referred toabove would indeed lead to some odd conclusions,namely: (i) one has no right to consider a countablyinfinite class of pairwise disjoint events the proba-bilities of which have the same order of magnitude;(ii) the limit of a sequence of probability distribu-tions need not be a probability distribution. Finally,he dwells upon the analogy between probability andmeasure supported by Frechet. De Finetti sharesFrechet’s opinion implying that each definition,even if of a mere mathematical nature, is more orless directly and distinctly triggered by intuition.Nevertheless, this very definition can effectively bearbitrary, provided that one confines himself to de-ducing purely formal conclusions from it. This is thecase of the definition of measure. A different case isconnected with the definition of weight, since “wecannot force a pair of scales to work according toour definition.” The case of probability is finally dif-ferent from the previous ones. If we, on the basisof a convention, state that P�Ak� = 0, k ≥ 1, en-tails P�∪∞k=1Ak� = 0, then we intuitively think of∪∞k=1Ak as a nearly impossible event, whereas theformal definition allows us only to conclude that 0is the value at ∪∞k=1Ak of the function which we,conventionally, have called “probability.”

Frechet promptly answered de Finetti’s new ar-guments (Frechet, 1930b) by expressing his sym-pathy for de Finetti’s position, at the same time, byshowing reluctance to remove complete additivity asa characteristic property for a probability distribu-tion. He resorts to some technical arguments thatde Finetti, in his subsequent answer (de Finetti,1930c), succeeds in using his own theory. See, for ex-ample, the argument employed by Frechet to showthat σ-additivity can be deduced from the “empiri-cal” definition of probability stated in Frechet andHalbwachs (1925).

We have lingered on the debate with Frechet justbecause in it de Finetti’s stance on probability foun-dations stands out most clearly. We are now alsoin a position to review the mathematical formula-

tion of his viewpoint, following Problemi determi-nati e indeterminati and Sul significato soggettivo(de Finetti, 1930d and 1931b, respectively). In pointof fact, in the corpus of de Finetti’s works the above-mentioned Fondamenti per la teoria delle proba-bilita is not listed. It was probably just a provisionaltitle for Sul significato soggettivo or for Problemi de-terminati e indeterminati, or for both.

A person who wants to summarize his degree ofbelief in the different values of a random event Eby a number p is supposed to accept any bet on Ewith gain c�p−1E�, where 1E denotes the indicatorof E and c is any real number chosen by an oppo-nent. Since c may be positive or negative, there isno advantage for the person in question in choos-ing a value p such that cp and c�p − 1� are bothstrictly positive for some c. Hence, p is an admis-sible evaluation of the probability of E if it meetsthe following principle of coherence: p has to ensurethat there is no choice of c so that the realizationsof c�p− 1E� are all strictly positive (or strictly neg-ative). The concept of coherence is then extended toa (finite or infinite) class E of events in the follow-ing way. The real-valued function P on E is saidto be a probability on E if, for any finite subclass�E1; : : : ;En� of E and any choice of �c1; : : : ; cn� inRn, n = 1;2; : : : ; the gain

G = G�E1; : : : ; Eny c1; : : : ; cn�

=n∑k=1

ck�P�Ek� − 1Ek�

is such that inf G ≤ 0 ≤ supG, where the infimumand supremum are taken over all constituents of�E1; : : : ;En�.

With regard to the problem of the existence ofat least a probability on a given class of events,de Finetti, in his early papers on the founda-tions of probability, confines himself to givingsome hints based on geometrical arguments. InSull’impostazione assiomatica, published in 1949,he dealt with this problem thoroughly by provingthe following:

Extension theorem. If E1 and E2 are classes ofevents such that E1 ⊂ E2; and if P1 is a probabilityon E1; then there is a probability P2 on E2 such thatP1 = P2 on E1.

De Finetti obtained this theorem independentlyof the almost contemporary Horn–Tarski extensiontheorem which, however, is less general than deFinetti’s; see Hornand Tarski (1948). De Finetti’sproof is by induction, transfinite induction when E2is not a countable class. Hence, his argument rests


on the axiom of the choice that he adopts here aswell as on other occasions, mentioning it explicitly.

Then, given a class E of events and an elementE of this class, any p in �0;1� represents a coher-ent assessment on E1 x= �E�, so that, in view of theextension theorem, there exists at least one proba-bility P on E such that P�E� = p.

After defining probability, de Finetti, in Sul signi-ficato soggettivo, proves that the usual rules of thecalculus of probability are necessary for the coher-ence of P on E. More precisely, he states that:

Theorem. If P is a probability on E; then:

(π1) A ∈ E⇒ P�A� ∈ �0;1�;(π2) � ∈ E⇒ P�� = 1;(π3) A1; : : : ;An ∈ E, ∪nk=1Ak ∈ E and Ak ∩Aj =

[ for 1 ≤ k 6= j ≤ n

⇒ P

( n⋃k=1

Ak

)=

n∑k=1

P�Ak�:

In particular, he deduces the following:

Characterization of P on an algebra. If E isan algebra, then P: E → R is a probability on E ifand only if P is nonnegative and finitely additive,with P�� = 1.

Hence, a probability coherent in de Finetti’s senseneed not satisfy the condition of σ-additivity. On theother hand, one obtains:

Coherence is preserved in a passage to thelimit. Let �Pn�n≥1 be a sequence of probabilities onE and let L = �E ∈ E: limn→+∞Pn�E� exists�. Then,if L 6= [; one has that P: L → R defined by P =limn→+∞Pn is a probability on L.

In connection with the principle of coherence letus mention that de Finetti’s paper Sul significatosoggettivo includes the first precise formulation ofa system of axioms of a purely qualitative natureconcerning the comparison between events. Such asystem of axioms defines a qualitative (or compara-tive) probability structure. In that very same paper,de Finetti considers the problem of the existence ofa quantitative subjective probability which agreeswith a given probability structure. De Finetti’s ap-proach to qualitative probability played an impor-tant role in Savage’s fundamental contribution tothe foundations of statistics; see Savage (1954).

Insofar as the concept of conditional probability isconcerned, de Finetti thinks of a conditional eventE�H as a logical entity which is true if E ∩H istrue, false if E is false and H is true, void if H isfalse (e.g., see de Finetti, 1937a). However, to thebest of our knowledge, he refrained from providing

a systematic theory of conditional probabilities. Hejust confined himself to mentioning the problem inTeoria delle Probabilita (de Finetti, 1970, volume 2;page 399 of the English translation), where he main-tains that:

it is necessary to strengthen the con-dition of coherence by saying that theevaluations conditional on H must turnout to be coherent conditional on H (i.e.,under the hypothesis that H turns outto be true : : :). Although this strengthen-ing of the condition of coherence mightseem obvious : : : there are several formsof strengthening conditions : : : :

Note. A concise review of the work coveringthe strengthening of the condition of coherence canbe found in Berti and Rigo (1989). We recall onlyone of the propositions following from the above-mentioned condition.

Characterization of conditional probabili-ties. Let E and H be two algebras of events suchthat H ⊂ E; and let P�·� · ·� be a real-valued func-tion defined on E ×H0; where H0 = H \ �[�. ThenP is a (coherent) conditional probability on E ×H0

if and only if:

(γ1) P�·�H� is a probability on E; for each H inH0;

(γ2) P�H�H� = 1 for all H in H0;(γ3) P�A ∩B�C� = P�A�B ∩ C�P�B�C� whenever

C and B ∩C belong to H0 and A and B are in E.

Conditions (γ1)–(γ3) are introduced in the formof axioms for conditional probability in de Finetti(1949). In view of the definition of conditional event,P�E�� must coincide with the probability of E, sothat we will write P�E� in place of P�E��. Hence,de Finetti’s theory of probability is a particular caseof the theory of coherent conditional probability.

In spite of the absence of a general theory,de Finetti, in Sulla proprieta conglomerativa(1930e), singled out an interesting circumstancewhich can occur in connection with a coherentconditional distribution: the absence of conglomer-ability. Let P�·� · ·� be a conditional probability onE × H0 according to the previous characterization,and let π = �H1;H2; : : :� be a denumerable parti-tion of � in H0. Given any E in E, if π is finite,then

P�E�� =∑k≥1

P�E�� ∩Hk�P�Hk��

�from �γ1� and �γ3��


and, consequently,

infkP�E�Hk� ≤ P�E� ≤ sup

k

P�E�Hk�:

De Finetti wonders whether this property, whichhe names conglomerability, holds even if π is in-finite. He gives a few examples which, contrary tointuition, show that P need not be conglomerable.The second example can be restated in the follow-ing terms. Let E be the power set of N2 and letH1 = ��n�×N; N×�m� x n ∈ N; m ∈ N�. Then, forany H in H1 and E in E, put

P1�E�H� = 0 or 1

according to E ∩H is finite or infinite.

In view of a coherence criterion due to Rigo (1988)(see Berti, Regazzini and Rigo, 1990 and 1996, forsome generalizations and applications), it is imme-diate to obtain that P1 is a coherent conditionalprobability on E ×H1. Hence, there exists at leastone extension of P1 to a conditional coherent prob-ability on E × E0, say P. Since, for E = ��n;m� ∈N2:n ≥m�,P1 �E��n� ×N� = P1�Ec�N× �n�� = 0 n ∈ N;

it is clear that P cannot be conglomerable on E andon Ec with respect to the partitions ��n�×N:n ∈ N�and �N × �n�:n ∈ N�, respectively. By the way, thesame example was later given by Levy (Levy, 1931b)and, in modern literature, it is known as Levy’sparadox. In de Finetti’s paper Sull’impostazione as-siomatica (1949), one can find an example of non-conglomerability with respect to a partition whoseelements have nonnull probabilities. Conglomerabil-ity is connected with disintegrability of probabilitymeasures. It has been studied in recent years byDubins and other authors (i.e., Dubins, 1975, andSchervish, Seidenfeld and Kadane, 1984). In mod-ern Bayesian literature, the phenomenon of noncon-glomerability has come up in the so-called marginal-ization paradoxes.

The above-mentioned paper Sull’impostazione as-siomatica (de Finetti, 1949), in addition to a com-plete account of subjective approach to probabilitybased on the principle of coherence, includes a de-tailed analysis of the differences between that ap-proach and the axiomatic theory of Kolmogorov.

A noteworthy analysis of the structure of afinitely additive probability is worked out in Lastruttura delle distribuzioni (de Finetti, 1955a),where de Finetti, independently of previous workscovering the same topic (Bochner, 1939; Sobczykand Hammer, 1944a, 1944b), provides a decom-position for finitely additive probabilities into thediscrete component (masses concentrated at the

points of a countable set), the agglutinated com-ponent (nonconcentrated in the previous sense,but concentrated on ultrafilters which constitutea countable class), and the continuous component(which can be indefinitely subdivided). Moreover,he decomposes the continuous component into thecondensed (singular) subcomponent and the dif-fuse (absolutely continuous) subcomponent, withrespect to a given continuous finitely additive mea-sure. Finally, he proves a Radon–Nikodym theoremfor finitely additive measures. Let us mention herethe method for proving these statements. The proofof the decomposition is based on a suitable coeffi-cient of divisibility, whereas the Radon–Nikodymtheorem is proved through an extension of thewell-known notion of Lorenz–Gini concentrationfunction.

Note. In recent years de Finetti’s extension ofthe notion of concentration function has been usedas a measure of robustness of Bayesian statisticalmethods and in order to obtain a new proof for “ex-act versions” of the Radon–Nikodym theorem; seeCifarelli and Regazzini (1987), Fortini and Ruggeri(1994) and Berti, Regazzini and Rigo (1992).

We conclude this subsection by explaining anotherimportant aspect of de Finetti’s contribution to thefoundations of probability: the treatment of the con-cept of expectation of a random quantity. The mainfeature of this treatment is that our author suggestsa definition on the basis of which the expectation ofa random quantity can be evaluated without assess-ing its probability distribution. An early hint of thisidea was included in a course of 1930 (de Finetti,1930f), but the first fully fledged attempt to makethis idea precise appeared later, in 1949. As to an ex-haustive exposition, it came up for publication onlyin 1970 with Teoria delle Probabilita. Starting fromthe concrete meaning of expectation of a randomquantity, de Finetti extends the betting scheme, re-called at the beginning of this present subsection,to random quantities, in order to define the conceptof prevision of a bounded random quantity. Hence,given any class K of bounded random quantitiesand a real-valued function P on K, one says thatP is a prevision on K if, for every finite subclass�X1; : : : ;Xn� of K and for every n-tuple �c1; : : : ; cn�of real numbers, one has

infn∑k=1

ck �P�Xk� −Xk�

≤ 0 ≤ supn∑k=1

ck �P�Xk� −Xk� :


In other words, P is a prevision if and only if thereis no (finite) betting system which makes uniformlystrictly negative the gain of the person who adoptsP as a system of unit-prices for bets on the elementsof K. It is clear that the probability of an event Emust coincide with the prevision of 1E and that thetheory of probability is included in that of previ-sion. The same argument used to prove de Finetti’sextension theorem can prove the following:

Proposition. If Ki; i = 1; 2 are classes ofbounded random quantities with K1 ⊂ K2; and ifP1 is a prevision on K1; then there is a prevision P2on K2 such that P1 = P2 on K1.

In view of this proposition, given any class K ofbounded random quantities there exists at least oneprevision on K. Moreover, if B designates the classof all bounded random quantities, then:

Proposition. The function P is a prevision onK ⊂ B if and only if P can be extended as a positive,linear functional P′ on B, such that inf X ≤ P′�X� ≤supX for each X in B.

This proposition shows that prevision meets allthe properties of the expectation, with the exceptionof the continuity property. Moreover, if one considerseach element of B as a bounded function from � intoR, then:

Proposition. The function P is a prevision onK ⊂ B if and only if there is a probability π onthe class of all subsets of � such that

P�X� =∫�Xdπ �X ∈ K�

with the integral thought of as an abstract Stieltjesintegral.

Apropos of the integral representation of a previ-sion, let’s also recall the paper, of a purely mathe-matical character, Sulla teoria astratta (de Finetti,1955b) as well as the paper Sull’integrale diStieltjes–Riemann (de Finetti, 1935a) written withM. Jacob. In the latter, our author deals with theproblem of the evaluation of P�X� when one knowsthe probability distribution function of X.

Note. In more recent years, de Finetti’s the-ory of prevision has been extended to conditionalbounded random variables (cf. Holzer, 1985, andRegazzini, 1985, for a review) and to random ele-ments taking values in a Banach space (cf. Berti,Regazzini and Rigo, 1994).

2.2 Processes with Independent Increments

In 1929, de Finetti started on new research re-garding functions with random increments. The cri-

sis of determinism and of the causality principle in-troduced a novelty into the scientific method. Thisbasically boils down to the replacement of classi-cal logic with probability calculus. Rigid laws whichstate that a certain fact is bound to occur in a cer-tain way are being replaced by probabilistic or sta-tistical laws stating that a certain fact can occurdepending on a variety of ways governed by proba-bility laws. De Finetti’s pioneer research on randomfunctions aimed at preparing an analytic appara-tus intended for the “translation” of deterministiclaws of physics into probabilistic laws. Then, givena scalar quantity whose temporal evolution is de-scribed by function X = X�λ�, λ ≥ 0, assume thatthe values taken by X�λ� are known for λ ≤ λ0 andconsider the (conditional) increment

�X�λ� −X�λ0��X�u�; u ≤ λ0�; λ > λ0:

Insofar as the probability distribution function φ�·�of such an increment is concerned, de Finetti con-siders the following cases:

(a) φ�·� is independent of X�u� for every u in�0; λ0�;

(b) φ�·� is independent of X�u� for every u in�0; λ0�;

(c) φ�·� dependent on the “history” of X on�0; λ0�.

This classification is inspired by Volterra’s classi-fication for deterministic laws of physics and deFinetti, following Volterra’s classification, calls φknown if case (a) occurs, differential if case (b)comes true and integral in case (c). Note thatKolmogorov’s attitude toward the role played bystochastic processes in the formulation of physicallaws was very close to de Finetti’s. See Kolmogorov(1931), where he deals with case (b). In Sulle fun-zioni a incremento aleatorio (de Finetti, 1929a),de Finetti deals with the problem of characteriz-ing the probability distribution of X�λ� in case (a).If X�0� ≡ 0 and φλ and ψλ denote the probabil-ity distribution function and characteristic functionof X�λ�, respectively, then �ψ1/n�·��n is the char-acteristic function of the sum of n independentincrements, identically distributed according to thelaw of X�1/n� −X�0�. If limn→+∞�ψ1/n�·��n exists,then de Finetti calls it the derivative law of the dis-tribution of X�λ� and denotes it by ψ′0�·�. Hence, iflogψλ�·� is differentiable with respect to λ, one has

ψ′0�·� = limn→+∞

�ψ1/n�·��n

= exp{∂

∂λlogψλ�·�

}∣∣∣∣λ=0


and

ψλ�·� = exp{∫ λ

0logψ∗ν�·�dν

}

with

ψ∗λ�·� = exp{∂

∂λlogψλ�·�

}:

In particular, if ψ∗λ is independent of λ (de Finetti,in this case, calls φλ known and fixed), one obtainsψ∗λ�·� = ψ

′0�·� and

ψλ�·� = exp�λ logψ′0�·�� = �ψ′0�·��λ;

which entails ψ1�·� = ψ′0�·� and

ψλ�·� = �ψ1�·��λ :In modern literature, these processes are knownas processes with homogeneous independent incre-ments, and ψ1 is called an infinitely decomposablecharacteristic function [indeed, ψ1 = �ψ1/n�n�. Atthis point, de Finetti considers the particular casein which

logψ′0�·� = imt−

σ2

2t2; m ∈ R; σ2 > 0;

which gives

ψλ� t� = exp{iλmt− λσ

2

2t2}

(i.e., X is a Brownian motion) and sketches a prooffor nowhere differentiability of X, a result generallyascribed to Paley, Wiener and Zygmund (1933).

In Sulla possibilita di valori eccezionali (deFinetti, 1929b), he considers processes with homo-geneous independent increments and shows thatψλ is continuous whenever X�λ� is continuous on�0;+∞� and X�λ� is different from cλ. The proofis based on an interesting geometrical argument.Moreover, the examples chosen to emphasize therelevance of the assumption of continuity for X arealso noteworthy: the Poisson process and the com-pound Poisson process. In fact, de Finetti’s methodenables him to deduce the characteristic propertiesof these processes in a way that is quite innovativewith respect to the original arguments propoundedby Lundberg (1903). Coming back to a continuousprocess X with known and fixed law, de Finetti,in Integrazione delle funzioni (de Finetti, 1929c),deals with the problem of determining the law ofthe integral

S =∫ λ

0X�u�du = lim

n→+∞Sn;

where

Sn =λ

n

n∑h=1

X

(h

nλ

):

In order to determine the probability distribution ofSn, he rewrites Sn as

Sn = λX�0� +λ

n

n∑h=1

�n− h+ 1�

·{X

(h

nλ

)−X

(h− 1n

λ

)}

and, from the properties of X, the characteristicfunction of Sn, ϕSn , satisfies

logϕSn�t � = ictλ+λ

n

n∑h=1

logψ1

(�n− h+ 1�λ

nt

)

when X�0� ≡ c. Hence,

limn→+∞

ϕSn� t� = ictλ+1t

∫ λt

0logψ1�u�du x= logϕ�t�:

In a σ-additive framework, this fact suffices to statethat ϕ is the characteristic function of S, while, inde Finetti’s finitely additive context, the previousargument states that �Sn�n≥1 has a limiting distri-bution but this need not be the distribution of S;see Section 2.1. This would be the case if �Sn�n≥1converged to S in probability. On the other hand, inorder to check on such a condition, we need some in-formation about the distribution of �S; Sn� for eachn. De Finetti, by way of example of the above result,states that the characteristic function of S equals

exp{it

(cλ+ m

2λ2)− σ

2λ3

3!t2}

when logψ1�t� = imt − �σ2/2�t2: This result is aprelude to analogous results stating the probabilitydistribution of particularly significant functionals ofthe Wiener–Levy process.

In two subsequent papers dated 1930 and 1931,he considered the problem of singling out allthe characteristic functions ψ such that ψλ is acharacteristic function for every λ > 0. In fact,this is equivalent to characterizing the charac-teristic function of X�λ�, X being any processwith homogeneous independent increments, andis equivalent to characterizing the class of all in-finitely decomposable characteristic functions. Inthe above-mentioned papers (de Finetti, 1930h,1931c), de Finetti obtains the well-known theorem:

Theorem. The class of infinitely decomposablelaws coincides with the class of distribution lim-its of finite convolutions of distributions of Poissontype.

Kolmogorov and Levy took these papers as astarting point for their renowned representationsfor infinitely decomposable characteristic functions;see Kolmogorov (1932a, b) and Levy (1934).


Apart from the brillant treatment included inthe second volume of Teoria delle Probabilita(de Finetti, 1970), de Finetti returned to the theoryof stochastic processes with independent incre-ments only once,with a short paper dated 1938,Funzioni aleatorie (de Finetti, 1938b). In this work,he assumes a critical attitude toward his previ-ous papers. In fact, in the above-mentioned papers,he considered the probability of transcendental(= infinitary) conditions while, in his opinion, onlyconditions depending on a finite number of values ofX�·� are susceptible of “concrete” probability eval-uations. The paper in question does not actuallyinclude new results, but rather a few suggestionsconducive, on one hand, to the notion of character-istic functional and, on the other hand, indicativeof a way to introduce the probability that X�·�satisfies certain conditions of an analytical na-ture. For instance, one cannot possibly speak ofthe probability distribution of X

′at the point t1,

but one can at least consider the distribution of�X�t1 + h� − X�t1��/h for each h. Consequently,one can consider the limit expression of that dis-tribution as h → 0 and interpret the limitingdistribution as an approximation of the law of thedifference quotient for h sufficiently small to justifythe approximation.

2.3 Sequences of ExchangeableRandom Variables

With regard to the connections between the sub-jectivistic viewpoint and the objectivistic one which,in a different way, characterizes the classical ap-proach (based on the notion of equally probablecases) and the frequentistic approach, de Finetticonsiders these approaches as procedures “thathave been thought to provide an objective mean-ing for probability,” but he immediately specifiesthat these same procedures are not necessarily con-ducive to the existence of an objective probability.On the contrary, by drawing that conclusion, oneencounters well-known difficulties, which do disap-pear only when one does not seek to eliminate but,on the contrary, one seeks to make the subjective el-ement more precise. Thus, the classical definition ofprobability, based on the relation of the number offavorable cases with the number of possible cases,can be justified immediately: indeed, if there is acomplete class on n incompatible events, and if theyare judged equally probable, then by virtue of thetheorem of total probability each of them will nec-essarily have the probability p = 1/n. The analysisof the frequentistic point of view is more complex.De Finetti proposed to break the analysis downinto two phases and explained their subjectivistic

foundations. The first phase deals with the rela-tions between evaluations of probabilities and theprevision of future frequencies; the second concernsthe relationship between the observation of pastfrequencies and the prevision of future frequencies.As regards the first phase, let us consider a se-quence of events E1, E2; : : : relative to a sequenceof trials and suppose that, under the hypothesisHN stating a certain result of the first N events, aperson considers equally probable the events EN+1,EN+2; : : : : Then, denoting by fHN

the prevision ofthe random relative frequency of occurrence of then events EN+1; : : : ;EN+n, conditional on HN, thewell-known properties of a prevision yield

pHN= fHN

;

where pHNindicates the probability of each EN+1,

EN+2; : : : conditional on HN. Hence, by estimatingfHN

via the observation of past frequencies, one ob-tains an evaluation of pHN

. But when is it permis-sible to estimate fHN

in such a manner? This is theproblem of the second phase. De Finetti’s answer is:when the events considered are supposed to be ele-ments of a stochastic process whose probability law,conditional on large samples, admits, as prevision ofthe future frequency, a value approximately equal tothe frequency observed in these samples. Since thechoice of the probability law governing the stochas-tic process is subjective, the prevision of a futurefrequency based on the observation of those past isnaturally subjective. De Finetti shows that the pro-cedure is perfectly admissible when the process isexchangeable, that is, when only information aboutthe number of successes and failures is relevant, ir-respective of exactly which trials are successes orfailures (cf. de Finetti, 1937a).

So far as we know, the notion of exchangeableevents dates to a communication by Jules Haag atthe International Mathematical Congress held inToronto (August 1924), published in 1928. In addi-tion to the definition, Haag exhibits an incompleteproof of the representation theorem. Independentlyof Haag’s work, de Finetti introduced exchange-able events as a probabilistic characterization of arandom phenomenon, that is, a phenomenon whichcan be repeatedly observed under homogeneousenvironmental conditions. He argued that a cor-rect probabilistic translation of such an empiricalcircumstance leads us to think of the probabilityof m successes and �n − m� failures in n trialsas invariant with respect to the order in whichsuccesses and failures alternate, whatever n andm may be. More precisely, in a communicationat the above-mentioned International Mathemati-


cal Congress held in Bologna, de Finetti defines asequence �En�n≥1 of events to be equivalent [nowa-days, the more expressive and unambiguous wordexchangeable, proposed by Polya (cf. de Finetti,1938c, 1939a) or Frechet (cf. de Finetti, 1976), isused; in fact, even the term symmetric, used byHewitt and Savage, admits ambiguity] if, for everyfinite permutation π, the probability distributionof �1E1

; : : : ;1En; : : :� is the same as the probability

distribution of �1Eπ�1�; : : : ;1Eπ�n�; : : :�. Contrary toHaag, de Finetti rigorously obtains the expressionfor the more general exchangeable distribution as aconsequence of the following:

Strong law of large numbers. If �En�n≥1 is asequence of exchangeable events, then

P

(sup

1≤p≤k

∣∣∣∣1n

n∑i=1

1Ei− 1n+ p

n+p∑i=1

1Ei

∣∣∣∣ ≤ ε)> 1− η

holds for every k = 1;2; : : : and for every strictlypositive ε; η; provided that n ≥N0 =N0�ε;η�:

In its turn, this result entails:

Theorem. If �En�n≥1 is a sequence of exchange-able events, then there is a probability distributionfunction F; whose support is included in �0;1�; suchthat

P

(1n

n∑i=1

1E i≤ x

)→ F�x�; n→+∞;

at each continuity point x of F.

Moreover, under the usual condition of completeadditivity for P, the first statement implies the ex-istence of a random variable θ such that

P

(supn≥m

∣∣∣∣1n

n∑i=1

1E1− θ

∣∣∣∣ ≤ ε)→ 1; m→+∞; ε > 0;

and F is the probability distribution function of θ.Starting from these basic results, one easily ob-

tains the following:

Representation theorem. If �En�n≥1 is a se-quence of exchangeable events, and if �x1; : : : ; xn� ∈�0;1�n; then

P�1E1= x1; : : : ;1En

= xn�

=N−n+sn∑M=sn

(N− nM− sn

)

(NM

) P

( N∑i=1

1Ei=M

)

where sn =∑ni=1 x i; n ≤N: Hence,

P�1E1= x1; : : : ;1En

= xn�

= limN→+∞

N−n+sn∑M=sn

(N− nM− sn

)

(NM

) P

( N∑i=1

1Ei=M

)

=∫�0;1�

tsn�1− t�n−sn dF�t �;

F being the limiting distribution of( N∑i=1

1Ei/N

)

N≥1:

The previous line of proof reproduces an argu-ment, recalled in La prevision (de Finetti, 1937a),due to Khinchin (1932b). As a matter of fact, deFinetti’s original argument in his communication atthe Bologna International Mathematical Congressand, more at length, in Funzione caratteristica(de Finetti, 1930g, 1932a), are of a merely analyticalnature (Fourier–Stieltjes transform). In any case,he was able to justify the evaluation of fHN

viapast frequencies, thanks to the the following:

Asymptotic theorem for predictive distribu-tions. If �En�n≥1 is a sequence of exchangeableevents, then

P

(sup

N≤n≤N+p

∣∣∣∣P�En+k�1E1; : : : ;1En

� −n∑i=1

1Ei

n

∣∣∣∣ ≤ ε)

> 1− ηfor every p and k in N and for every strictly positiveε and η; provided that N ≥N0�ε;η�.

In three short Notes published in Atti della RealeAccademia dei Lincei, in 1933, de Finetti extendedthe previous statements to sequences of exchange-able random quantities. In particular, he extendedthe strong law of large numbers to exchangeablerandom variables with finite fourth moment and heexplained an argument to extend the representationtheorem. Apropos of this latter point, further devel-opments are explained in de Finetti (1937a), wherethe representation theorem is formulated in the fol-lowing terms.

Representation theorem. Let �Xn�n≥1 be a se-quence of exchangeable random variables and let F

denote the class of the probability distribution func-tions on R equipped with the Levy metric. Then thereis a probability µ on the class of all subsets of F suchthat

P�X1 ≤ x1; : : : ;Xn ≤ xn� =∫

F

n∏k=1

V�xk�µ�dV�

for every xk in R; k = 1; : : : ; n; n ∈ N.


It has to be stressed that in de Finetti’s formu-lation of the representation theorem, in conformitywith the principle of coherence, P is thought of asfinitely additive. Hence, that very same formulationneed not be equivalent to the usual proposition “theXn’s are exchangeable if and only if they are con-ditionally i.i.d., given a random quantity θ,” whichholds if P is σ-additive.

Note. This form of de Finetti’s theorem is quitewidely known. It has been extended to random el-ements with values in a compact Hausdorff spaceby Hewitt and Savage (1955) and to random ele-ments with values in a standard Borel space byAldous (1985). On the other hand, Dubins andFreedman (1979) give an example to show thatthe usual formulation is false for general randomelements; see also Freedman (1980) and Dubins(1983). Buhlmann (1960) found the appropriaterepresentation for processes with exchangeable in-crements and Freedman (1962a, b, 1963) began towork on more general types of symmetry. Freed-man (1962a) and Diaconis and Freedman (1980a)dealt with the characterization of discrete chainswith a law which is invariant with respect to theorder of transitions. In the first paper, one provesthat, if the chain is stationary, then it must bea mixture of Markov chains. In the second pa-per, the authors reach the very same conclusionunder a suitable condition of recurrence. Daboni(1975, 1982) obtained characterizations of mix-tures of particular Markov processes starting fromexchangeability of interarrivals times, and usedsuch a characterization to prove two well-knownHausdorff and Bernstein theorems concerning com-pletely monotone functions. Significant extensionsof the above-mentioned results to general statespaces, and/or to continuous time, can be found inFreedman (1984) and Kallenberg (1973, 1974, 1975,1982).

According to Diaconis (1988):

de Finetti’s theorem presents a real-valued exchangeable process as a mix-ture over the set all measures. Most ofus find it hard to meaningfully quan-tify a prior distribution over so largea space. There has been a search foradditional restrictions that get thingsdown to a mixture of familiar fami-lies parametrized by low dimensionalEuclidean parameter spaces.

In this setting, in addition to Freedman (1963),let us mention Dawid’s characterization of themean-zero p-dimensional covariance mixture of

normal distributions and Smith’s characterizationof location–scale mixtures of univariate normaldistributions (cf. Dawid, 1978 and Smith, 1981).The multidimensional extension of Smith’s theoremhas been given by Diaconis, Eaton and Lauritzen(1992). These results characterize those modelsthat are compatible with a certain choice of datareduction. This, in turn, involves sufficiency and islinked to de Finetti’s reconstruction of the Bayes–Laplace paradigm; see Section 3. Diaconis andFreedman (1984) provide a framework for char-acterizing mixtures of processes in terms of theirsymmetry properties and sufficient statistics. Asan application, mixtures of the following kinds ofprocesses are characterized: coin-tossing processes(de Finetti); sequences of independent identicallydistributed normal variables; sequences of inde-pendent, identically distributed integer-valuedgeneralized exponential variables. Extensions ofthis last characterization are included in Kuchlerand Lauritzen (1989) and Diaconis and Freedman(1990). Other interesting de Finetti style theoremscan be found in Lauritzen (1975), Ressel (1985),Accardi and Lu (1993) and Bjork and Johansson(1993). To conclude this short review of recentliterature related to de Finetti’s representation the-orem, let us mention central limit theorems: byBuhlmann (1958), by Blum, Chernoff, Rosenblattand Teicher (1958), by Klass and Teicher (1987) andby Eaton, Fortini and Regazzini (1993). The limit-ing distribution of

∑n1 X i/n has been determined

by Cifarelli and Regazzini (1990, 1993) under theassumption that µ is a Ferguson–Dirichlet distribu-tion. Recently, Berti and Rigo (1994) have provideda Glivenko–Cantelli style generalization of theasymptotic theorem for predictive distributions.

During his subsequent scientific career, de Finettiwent back to exchangeability on two occasions: first,to deal with the so-called degenerate sequences ofexchangeable random events; second, to considerthe problem of extending finite exchangeable se-quences. In Gli eventi equivalenti e il caso degenere(1952), he recalls that the En’s are exchangeable ifand only if

P�En+1�1E1= x1; : : : ;1En

= xn� = p�n�( n∑

1

xk

)

for every n and �x1; : : : ; xn� in �0;1�n, and

p�n�( n∑

1

xk

)qn+1

( n∑1

xk + 1)

= q�n�( n∑

1

xk

)pn+1

( n∑1

xk

)


where q�n� = 1− p�n�, and he designates as de-generate any exchangeable distribution such thatp�n�r = 0 or 1, for some r and n. Then, after some

interesting considerations of a geometrical nature,he considers the problem of representing a de-generate exchangeable sequence and connects thisproblem with the use of improper priors for theBayesian analysis of the Bernoulli parameter; seealso Daboni (1953). As for the problem of extend-ing sequences of exchangeable random events, itis well known that a finite exchangeable processmay or may not be the initial segment of someother exchangeable sequence of more steps. Then,in Sulla proseguibilita (1969), de Finetti suggests ageometric approach to deal with the following prob-lem: is a given n-exchangeable process extendible?In the affirmative, how many steps can it be ex-tended for, while preserving exchangeability? If thegiven n-exchangeable process is r-extendible, canwe characterize the r-exchangeable process withrespect to which the given process is the initialn-segment?

Note. De Finetti’s geometric viewpoint was fol-lowed by Crisma (1971, 1982), Spizzichino (1982)and Wood (1992). A very clear exposition of suchapproach is given in Diaconis (1977).

In a communication at the Colloque de Genevein 1937, Sur la condition d’equivalence partielle(de Finetti, 1938a), de Finetti proposes the follow-ing extension of the notion of exchangeability. Heconsiders k sequences of random events �E�i�n �n≥1,i = 1; : : : ; k, and he defines the resulting arrayof events to be partially exchangeable if the prob-ability law of that array is the same as the lawof ��E�1�π1�n��n≥1; : : : ; �E

�k�πk�n��n≥1 �, where π1; : : : ; πk

are arbitrary finite permutations. After explainingthe meaning of this condition in connection withthe inductive problem, he states the representationtheorem for the most general law for an array ofpartially exchangeable events.

Note. In more recent years, finite exchange-able sequences have been studied by Diaconis andFreedman (1980b), who analyzed the behavior ofthe variation distance between the distribution ofan n-exchangeable process and the closest mix-ture of i.i.d. random variables. Those very sameauthors give similar theorems for particular ver-sions of de Finetti’s theorem (like normal location,or scale parameters, mixtures of Poisson, geomet-ric and gamma etc.) and for exponential families;see Diaconis and Freedman (1987, 1988). Diaco-nis, Eaton and Lauritzen (1992) consider a finite

sequence of random vectors and give symmetry con-ditions on the joint distribution which imply that itis well approximated by a mixture of normal distri-butions.

As far as partial exchangeability is concerned,let us recall that contemporary scholars tend touse that term in a broad sense; see Aldous (1981)and Diaconis and Freedman (1980a). In the latterpaper, for instance, partial exchangeability des-ignates invariance with respect to the order oftransitions. As a matter of fact, Petris and Regazz-ini (1994), following de Finetti (1959), analyze thestrict relationships between that invariance andde Finetti’s partial exchangeability of the array of“subsequent states;” see also Zabell (1995). Par-tial exchangeability, in the sense of Diaconis andFreedman, for finite sequences has been consideredby Zaman (1984, 1986). Central limit theorems forpartially exchangeable arrays are proved in For-tini, Ladelli and Regazzini (1994). Von Plato (1991)and Scarsini and Verdicchio (1993) have dealt withthe extendibility of finite partially exchangeablerandom elements.

Finally, let us mention that in April 1981 anInternational Conference on “Exchangeability inProbability and Statistics” was held in Rome. Theproceedings of the conference, edited by Koch andSpizzichino, include 8 general lectures and 21 in-vited contributions, in addition to 2 articles by deFinetti and Furst, respectively.

A full survey of classical work on exchangeabilitycan be found in Aldous (1985). Diaconis (1988) canbe thought of as a useful updating supplement tothe Aldous monograph.

2.4 Other Contributions to Probability Theoryand Its Applications

We devote this subsection to four papers ofde Finetti for which it would be difficult to findsuitable pigeon holes. The first, Resoconto criticodel colloquio (de Finetti, 1938c), represents an ac-count of a conference, held in Geneva, October 1937,and coordinated by Frechet, devoted to the theoryof probability. The author provides a concise butclear-cut summary of each of the 16 lectures actu-ally delivered at the Geneva conference, reportingin addition both the discussions originated by eachlecture and some personal remarks. As a result,we are presented with a stimulating picture of thestate of probability theory at the end of a partic-ularly productive period. We list the lecturers andthe subjects dealt with in Appendix 1.


The second paper we want to draw to your at-tention is La teoria del rischio (de Finetti, 1939b),which deals with the risk theory from Lundberg’sstandpoint. We mention it in the present sectionbecause of its interesting connections with a fun-damental identity later proved by Wald; see Wald(1944), Section 3. Assume that G1 and G2 arestrictly positive integers, and consider two playerswho start with bankrolls G1 and G2, respectively.Then consider a sequence of random variables�ξn�n≥1 with their ranges in �−1;1�: if ξ i = 1 (−1,respectively), we suppose that the second playerpays one unit to the first (the first pays the second,respectively). Then Sk x= ξ1 + · · · + ξk can be inter-preted as the amount won by the first player fromthe second after k turns. If

τ = inf �n ≥ 1:Sn ≤ −G1 or Sn ≥ G2� ;

then P1 = P�S τ = −G1� [P2 = P�S τ = G2�, respec-tively] represents the probability of ruin for the firstplayer (the second, respectively) and, under suitableconditions of fairness, one has

P1 +P2 = 1; P2G2 −P1G1 = 0:

In particular, these conditions are satisfied if theξ’s are independent and identically distributed withP�ξ1 = 1� = P�ξ1 = −1� = 1/2. As a matter offact, de Finetti considers more general situations inwhich one, for example, has E�ξn+1�ξ1; : : : ; ξn� = 0for every n ≥ 1, and he deals with the problem ofdetermining the probability of ruin when the gameis unfair, by resorting to a trick which dates backto De Moivre; see De Moivre (1711) and Thatcher(1970). Assume that the ξ’s are independent andidentically distributed with P�ξ1 = a� = p in �0;1�and P�ξ1 = b� = q = 1 − p: Then attach value�exp�αξi� − 1� to the gain ξ i and choose α in sucha way that the resulting gain is fair that is, chooseα0 6= 0 for which E�exp�α0ξi� − 1� = 0. Afterwards,de Finetti proves that the sign of α0 is opposite tothe sign of E�ξi� and that E�exp�α0

∑ni=1 ξi�� = 1

for every n. Hence, from the well-known formulafor fair games

P1�exp�−α0G1� − 1� +P2�exp�α0G2� − 1� = 0;

P1 +P2 = 1;

one has

P1 = exp�α0G1�exp�α0G2� − 1

exp�α0�G1 +G2�� − 1;

E�exp�α0Sτ�� = 1:

Our author dwells upon the meaning of α0 as a “in-dex of risk” when E�ξ1� > 0 (α0 < 0) and G2 = +∞

(the case, e.g., of an insurance company). In thiscase, one can write

P1 = exp�−G1/B�; B = −1/α0;

and B represents the value of the company’s initialcapital for which the probability of ruin equals 1/e,so that de Finetti calls it riskiness level.

In Trasformazione di numeri aleatori (de Finetti,1953a), he proves that, given two continuous andproper probability distribution functions on R, sayF and G, there is a random variable on R with itsrange in �0;1� whose probability distribution func-tion is uniform on �0;1� both under F and underG. This implies, for instance, that two statisticians,with different continuous posterior distributionsfor a real-valued parameter, can still find a com-mon ground on which their respective opinions docoincide.

To conclude, we refer to the subject matter of thefirst paper taken into consideration in the presentsection: convergence of sequences of random vari-ables. As mentioned in Section 1.3, the GIIA, in1933, published a paper in which Glivenko provedthe almost-sure convergence of supx∈R �Fn�x�−F�x��to 0 (as n → +∞), where Fn represents the em-pirical distribution function associated with a ran-dom sample from the continuous probability distri-bution function F. The extension of Glivenko’s the-orem to whichever probability distribution functionF is generally ascribed to Cantelli. In fact, in apaper published the same year in GIIA, Cantelliclaims that the problem of possible discontinuitypoints of F can be dealt with by resorting to thedefinition of random variable he had proposed eversince 1916. In point of fact, according to Cantelli,such a definition ought to imply that the jump atx of F represents the probability concentrated atx. Actually, it is not clear how a definition of ran-dom variable could produce the above phenomenonwhich, instead, has to be ascribed to the propertiesof the probability distribution of each element of therandom sample. For example, if this distribution isσ-additive on the Borel class in R, then any jump ofF takes on just that meaning, but this is not neces-sarily true in the finitely additive case. The correctinterpretation of the jumps of F in connection withthe so-called fundamental theorem of mathemati-cal statistics is included in a paper by de Finetti,Sull’approssimazione empirica (de Finetti, 1933b),where the above-mentioned extension of Glivenko’stheorem is thoroughly stated. This paper, which pre-cedes Cantelli’s, includes also some interesting hintsat how to investigate that very same problem in thecase of the existence of possible adherent masses ata discontinuity point of the distribution function. In


a nutshell, de Finetti suggests that, in order to ob-tain a proposition whose validity is independent ofthe nature of the jumps of F, one should use Levy’smetric as a substitute for the uniform metric. More-over, he replaces Glivenko’s original argument witha simpler procedure which makes the most of Can-telli’s strong law of large numbers in Bernoulli tri-als. Therefore, even if research on the convergenceof the empirical distribution function, in the early1930s drew inspiration from Cantelli’s works on con-vergence of random sequences, the “fundamentaltheorem of mathematical statistics” ought to be as-cribed, strictly speaking, to Glivenko and de Finetti.

3. DE FINETTI’S WORK IN STATISTICS

For expository convenience, we will split deFinetti’s contributions to statistics into two subsec-tions: the former devoted to description of frequencydistributions, the latter to inductive reasoning.

3.1 Description of Frequency Distributions

The early de Finetti paper in this field deals withGini’s mean difference. In Calcolo della differenzamedia (de Finetti, 1930i), de Finetti and Paciellostated the identity

�mean difference =�∫

R2�x− y�dF�x�dF�y�

= 2∫

RF�x��1−F�x��dx;

and, subsequently, in Sui metodi proposti (1931d),de Finetti discussed different methods for the com-putation of the mean difference by showing the su-periority of the so-called de Finetti–Paciello method.

That very same year saw the publication of oneof the most important papers by de Finetti in thefield of the analysis of statistical data: Sul concettodi media (de Finetti, 1931e). The starting pointof this new investigation was a definition of meangiven two years before by Chisini, an Italian distin-guished geometer who was de Finetti’s teacher atthe University of Milan. According to such a defi-nition, given n quantities x1; : : : ; xn the concept ofmean of x1; : : : ; xn is meaningful and operationallydetermined when there is a new quantity z whichdepends on x1; : : : ; xn,

z = fn�x1; : : : ; xn�;so that one can think of a mean as a value x which,if each xi is replaced by x, does not alter the valueof z at �x1; : : : ; xn�. As an example, assume that thenatural increase in a human population, in threeconsecutive years, is 1:2%, 0:2% and 0:8%. What isthe mean increase in that triennium? If z is taken to

be the total increase in the population of the sameperiod, then the mean x has to satisfy

1:012 · 1:002 · 1:008 = �1+ x�3;

which yields x ' 0:00729. De Finetti extendsChisini’s definition of a mean to distribution func-tions: given a class F of frequency (or probability)distribution functions on R and a real-valued func-tion f on F, a mean of F in F, with respect to theevaluation of f, is any number ξ = ξ�F� such that

f�F� = f�Dξ�;

where Dx denotes the probability distributionfunction which degenerates at x. Assume thatF = F�A;B� is the class of all distribution func-tions whose support is included in �A;B� with−∞ < A < B < +∞, and that m: F → R is de-fined by f�Dm�F�� = f�F� for every F in F. In otherwords, m is a mean on F�A;B� with respect to theevaluation of f. De Finetti considers three prop-erties which, though they might be optional, areanyway significant for a general mean with respectto some specific classes of problems. Thus, he saysthat m is reflexive if m�Dx� = x; m is strictly in-creasing if, for any pair of elements of F, say Fand G, such that F�x� ≤ G�x� for every x in R

and F 6= G, one has m�F� > m�G�; m is associa-tive if for every t in �0;1� and F, G, F∗, G∗ suchthat m�F� = m�F∗� and m�G� = m�G∗�, one hasm�tF+�1− t�G� =m�tF∗+�1− t�G∗�. The meaningof the second property is clear when F denotes theclass of all random gains taking values in �A;B�.Indeed, in that case, F�x� ≤ G�x� for every x in R

and F 6= G means that, for each x in R, the proba-bility under F of a gain greater than x is not lowerthan the probability of the same event under G (itis indeed greater for some x). Nowadays, this con-dition is identified with a well-known definition ofstochastic dominance. Finally, associativity statesthat a mean remains unchanged in spite of changesin some part of the distribution, provided these donot alter the corresponding partial mean. In con-nection with these properties, de Finetti proves asignificant extension of a theorem independentlyproved by Kolmogorov (1930) and Nagumo (1930):

Representation theorem for an associativemean (de Finetti–Kolmogorov–Nagumo]. Sup-pose that m: F�A;B� → R is a reflexive, strictlyincreasing and associative mean. Then there is afunction φ, continuous and strictly increasing in theclosed interval �A;B�, for which

�∗� m�F� = φ−1(∫

Rφ�x�dF�x�

); F ∈ F�A;B�:


Moreover, φ is uniquely determined up to lineartransformations. Conversely, if m is defined by �∗�;for a φ with the properties stated, then it satisfiesreflexivity, strict monotonicity and associativity.

This formulation is drawn from the classic bookby Hardy, Littlewood and Polya (1934), who followthe guidelines of de Finetti’s original proof. The 1931paper includes a differential criterion which provesof interest in stating whether a mean is associa-tive. Moreover, denoting mean (*) by mφ, de Finettiproves that mφ1

≥ mφ2on F�A;B� if and only if

φ2 ◦φ−11 is convex. Finally, he characterizes the mφ

which are homogeneous or translative. A mean mφ

is said to be homogeneous (translative, respectively)ifmφ�Fa� = a·mφ�F� for every a > 0 [mφ�Fa� = a+mφ�F� for every a, respectively] and F in F�A;B�,with Fa�x� = F�x/a� [Fa�x� = F�x − a�, respec-tively] for every x in R.

Characterization of homogeneous and trans-lative means. mφ is homogeneous if and only if φis any linear transformation of φ1�x� = xm, m ∈R \ �0�, or of φ2�x� = log x. Correspondingly, mφ istranslative if and only if φ is any linear transforma-tion of φ3�x� = exp�cx�; c ∈ R\�0�; or of φ4�x� = x.Therefore, the arithmetic mean is the sole mφ meanwhich is both homogeneous and translative.

The paper A proposito di correlazione (de Finetti,1937b) originates from a debate on the use and mis-use of the correlation coefficient which came to thefore during the 23rd Session of the InternationalStatistical Institute (ISI) (London, 1934). De Finetti,thanks to his in-depth knowledge of metric and, inparticular, pre-Hilbert spaces, deals with correlationfrom a geometrical viewpoint. First, he considers theset of all random quantities X, with E�X2� < +∞,as a metric space with distance between X and Ygiven by the standard deviation of �X−Y�. In sucha case, X and Y are thought of as coincident if andonly if P��Y − X − a� > ε� = 0 for every ε > 0and for some constant a. This metric space can beviewed as a pre-Hilbert space with inner productσ�X�σ�Y�r�X;Y�, where σ denotes standard devi-ation and r correlation coefficient. After stressingsome interesting descriptive properties on the ba-sis of geometrical properties of the space in ques-tion, de Finetti deals with some special cases andtackles the problem raised in the ISI debate: canthe correlation coefficient be considered as a mea-sure of monotone dependence between two numeri-cal properties of a population? The answer is posi-tive if and only if the marginals F1 and F2 are suchthat F1�x� = F2�ax+ b� = 1−F2�−ax+ β� for ev-ery x and for some a > 0 and b, β in R. Finally, in

connection with the problem of defining true mea-sures of monotone dependence, de Finetti proposesthe following two indices for a bivariate probabilitydistribution function with density function ϕ:

c1 =∫0ϕ�x;y�ϕ�ξ;η�dxdydξdη− 1/2;

c2 = 2σ�X�σ�Y�r�X;Y�;with 0 = ��x;y; ξ; η� ∈ R4: �ξ−x��η−y� > 0�. Afterclarifying that c1 and c2 may have opposite signs,de Finetti mentions an analogy between c1 and c2:c1 takes into consideration the signs of �ξ−x��η−y�only, while c2 considers the value of that product,too. Clearly, c2 is the covariance of ϕ up to a mul-tiplicative constant, while c1, surprisingly, is thewell-known Kendall’s τ; see Kendall (1938). Hence,de Finetti introduced this index into statistical lit-erature one year before Kendall did, so that onemight designate it as de Finetti–Kendall index (ofmonotone dependence). To tell the truth, the Ency-clopedia of Statistical Sciences informs us that thisindex had already been discussed around 1900 byFechner, Lipps, and Deuchler and more theoreticallyin the 1920s by Esscher and Lindeberg.

De Finetti’s line of reasoning in descriptive statis-tics is explained in La matematica nelle concezionistatistiche (de Finetti, 1943), where he introducesa natural distinction with reference to the descrip-tive power of means and statistical indices with re-spect to concrete problems. In the light of the abovedistinction, these indices have a significant valueif they embody the actual statement of a problem;otherwise they have a mere indicative value whenthere is no given stated problem to be answered.For example, a mean has significant value if it is as-sessed according to Chisini’s definition since, in thiscase, it meets the precise requirement of a rationalstatement. Passing from means to indices, de Finetticlaims that Chisini’s line of reasoning relating tomeans can be extended to other statistical indicesin order to obtain significant information with refer-ence to important aspects of a distribution. He givessome hints at a treatment for large classes of vari-ability indices, and quotes his paper on the correla-tion coefficient. Finally, he mentions an interestingaspect of the construction of statistical indices incase of indices with a mere indicative value. He, in-deed, points out that some important properties of adistribution have a more or less precise qualitativemeaning which allows us to set up a partial order-ing in the class of all distributions. For instance,the usual concept of concentration of a positive andtransferable character leads us to state that distri-bution F is no more concentrated than distributionG when the (Lorenz–Gini) concentration curve of F


lies above the concentration curve of G. Then, twoelements are not comparable if their concentrationcurves intersect. Generally speaking, any statisti-cal index defines a complete ordering in the classof all distributions so that, if a “natural” partial or-dering exists as a consequence of the definition ofthe property taken into consideration, then an in-dex ought to agree with such a partial ordering. Ina previous paper (de Finetti, 1939c), de Finetti hadlinked these ideas to lattice theory, as explainedin a booklet by Glivenko; see Glivenko (1938). Itseems to us that today’s theory of stochastic dom-inance, maybe unconsciously, is very much imbuedwith principles clearly reminiscent of de Finetti’sviewpoint on the description of frequency distribu-tions. A course length treatment which develops thisapproach can be found in Regazzini (1987).

We conclude this subsection by mentioning afurther contribution to descriptive statistics byde Finetti: the extension of Levy’s concept of dis-persion to multivariate distributions; see de Finetti(1953b).

Despite his interest in descriptive statistics, onseveral occasions de Finetti doubted the very valid-ity of any statistical argument taken disjointly frominduction. In his opinion, correct inductive proce-dures are necessary in order that descriptive meth-ods may rationally embody all available informationwith respect to the purpose of a given statisticaldata analysis. Hence, it is high time indeed we re-viewed his work in the field of statistical inference.

3.2 Induction and Statistics

Before getting to the heart of the matter, that is,de Finetti’s outlook on statistical inference, let usmention some specific cases considered by our au-thor. During his stay at the Istituto Centrale di Sta-tistica, he worked together with Gini on the prospec-tive growth of the Italian population and, on thatoccasion, he dealt with the problem of determiningthe parameters of a logistic curve. The conclusionsdrawn from this study can be found in de Finetti(1931f ).

In Sulla legge di probabilita degli estremi(de Finetti, 1932b), he considers the problemof determining the distribution of Mn = max�X1; : : : ;Xn� when X1; : : : ;Xn are i.i.d. randomvariables. In particular, he deals with the problemof “estimating” Mn by the median ξn of the proba-bility distribution function of Mn. If the probabilitydistribution function F of each Xi is continuousand strictly increasing, then ξn is the unique rootof the equation F�ξn� = 1/ n

√2. So, after provid-

ing the condition under which �Mn − ξn� → 0 inprobability (as n → +∞), he states that ξn can be

considered as a good estimator of Mn, for n suf-ficiently large. In order to evaluate the order ofmagnitude of �Mn − ξn�, he considers the case inwhich F is normal and proves that ξn�Mn − ξn� hasa limiting distribution F0 (as n → +∞), precisely:F0�x� = �1/2�exp�−x�; x > 0. Some specialists in thefield of order statistics acknowledge the pioneernature of de Finetti’s paper; see David (1981).

In a very short communication at the 23rd Meet-ing of the Societa Italiana per il Progresso delleScienze (Naples, 1934), he tackled the problemwhich is now known as nonparametric estimationof a cumulative distribution function; see de Finetti(1935b). In point of fact, what we have is merely astatement of the problem within a proper Bayesianframework. On one hand, he assumes that φ:Rn+1 → R represents a density function for thelaw of the random vector �X1; : : : ;Xn;X� andidentifies the problem at issue with the one of de-termining the distribution of X, after observingX1 = x1; : : : ;Xn = xn. Apropos of this solution,he recalls that, if �X1; : : : ;Xn;X� is the initialsegment of a sequence of exchangeable randomvariables, then the cumulative distribution corre-sponding to φ is close to the empirical distributionof the xi’s, provided that n is sufficiently large. Onthe other hand, de Finetti thinks of the “unknown”distribution as a realization of a random func-tion whose trajectories are probability distributionfunctions. Hence, after assigning the probabilitydistribution of that random function and the con-ditional probability law of data given that verysame distribution function, one will use the Bayestheorem to obtain the posterior distribution of therandom function. Within this framework, the un-known distribution is estimated by some suitable“typical value” of the posterior distribution (mean,median, etc.). In our opinion, the latter of the twoabove-mentioned approaches is valuable on two ac-counts. In the first place, by trying another tack,he clearly envisaged the Bayesian nonparametricanalysis of statistical problems: please note thatin statistical literature this came into the fore 40years later, thanks to Ferguson (1973). In the sec-ond place, in the paper in question, de Finetti’sview stood out most clearly.

3.2.1 Probability and induction. Anyway, thisview is thoroughly expounded in La prevision(de Finetti, 1937a) and in La probabilita e lastatistica (de Finetti, 1959). Let us then confineourselves to reporting only the most significant as-pects. In spite of the long lapse of time betweenthe two works, the basic train of thought expressedthroughout is always quite consistent. True enough,


the 1959 lectures do include also a critical surveyof the major stances of the time on statistical in-ference, but in 1937 all the “core” had already beenclearly formulated.

According to de Finetti’s interpretation, statisti-cal inference is a special case of reasoning by induc-tion, where “reasoning by induction” means learningfrom experience. However:

to speak of inductive reasoning means toattribute a certain validity to this wayof learning, to view it not as an upshotof quirky psychological reaction, but as amental process in its own right suscepti-ble of analysis, interpretation and justi-fication. When it comes to induction, thetendency to overestimate reason, oftento the point of excluding tout court anyother factor, can prove a very detrimen-tal bias. Reason is invaluable as a sup-plement to other intuitive faculties, butshould never be a substitute for them : : : :

A consequence of this bias is the eleva-tion of deductive reasoning to the statusof a standard, though actually all non-tautological truths are based on some-thing else. Thus inductive reasoning isgenerally considered as something on alower level, warranting caution and sus-picion. Worse still, attempts to give it dig-nity try to change its nature making itseem like something that could almost beincluded under deductive reasoning.

In point of fact, there are often at-tempts to explain induction withouteven introducing the term “probability”,or else trying to wrest the term fromits every-day significance as a measureof a degree of belief attributed to thevarious possible alternatives. [From theEnglish translation of La probabilita ela statistica in de Finetti, 1972, pages147–148.]

It is what routinely occurs in all statistical fre-quentistic procedures, when the very nature andmeaning of probabilistic reasoning is indeed basi-cally altered, for example: attempting to expressits conclusions as predictions, and speaking of suchthings as the possibility of verification or of agree-ment with experience; speaking of the “impossibil-ity” of highly improbable events; and resorting insome way to those interpretations which many con-sider necessary in order to give status to the cal-culus of probability by suppressing all trace of itspeculiar task, which is to deal with uncertainty. In

more general terms, by postulating the existence ofnatural laws with more or less narrowly predeter-mined characteristics, you can even pass off induc-tive reasoning as actually deductive. All the concep-tions which view the determinism as the necessarypostulate of science itself are basically founded onthis assumption, and, consequently, they reject thevery core of inductive reasoning.

With these considerations as a starting point,de Finetti proceeds very much along the lines ofHume’s idea of causality. When we regard anyevents as causally connected, all we do and canobserve is that they frequently and uniformly go to-gether, and the impression or idea of the one eventbrings with it the idea of the other. A habitual asso-ciation is set up in the mind, and as in other formsof habit, so in this one, the working of the associ-ation is felt as compulsion. This feeling is the onlydiscoverable impressionable source of the idea ofcausality. In other words, when we assert a causalconnection between any two objects, the sole ex-perience or evidenced necessitation is not in thembut in the habituated mind. That is where mathe-matics comes in handy: just to make precise whatcan indeed be made precise. As a matter of fact,in mathematics, too, inductive reasoning cannot beruled out. Far from it, in the creative moment itdoes prove necessary. Nobody could set out to provea theorem if this very theorem could not have somekind of likelihood. Levy used to say that if youwant to get somewhere, you must anyway envis-age your destination (insight) before starting off onyour actual journey (logic).

Let us now set aside all the considerations men-tioned in passing to concentrate on the role of math-ematics in the formulation of inductive reasoning.In order to tackle induction from the mathematicalangle it is imperative one should secure a mathe-matical instrument suitable for mastering the logicof uncertainty. Such an instrument does exist, andit is the theory of probability viewed in a broadsense as in de Finetti’s subjectivistic approach. Ac-cording to de Finetti, the role of probability withinthe framework of inductive logic boils down to thefollowing: pinpointing how the evaluation of proba-bility relative to future events “must change” follow-ing the result of events actually observed. The term“must change” is not to be interpreted as “must becorrected”. The probability of occurrence of a condi-tional event is nothing but a coherent extension ofa given probability law and not a “better evaluationof the probability of an unconditional event” (cf. deFinetti, 1970, Chapter 4). In any case, this “change,”properly interpreted, is the mathematical equiva-lent of “learning by experience.” Consequently, in-


ductive logic boils down either to the principle ofcompound probabilities or to its more sophisticatedvariant which passes under the name of the theo-rem of Bayes.

In this sense, de Finetti is Bayesian even if, apro-pos of this qualification, some cautions are in orderboth in connection with the original Bayes-Laplaceparadigm and with respect to the neo-Bayesianpractice that paradigm is imbued with. Indeed, inLa prevision he strongly criticizes the usual inter-pretation of the Bayes–Laplace paradigm and, inLa Probabilita e la statistica, he does not mincewords and speaks of “reconstruction of the clas-sical formulation according to the subjectivisticviewpoint” with reference to the introduction ofexchangeability and partial exchangeability intoinductive logic.

A first circumstance to be mentioned here regardsthe nature of the condition of exchangeability, inother words how to translate the empirical vagueidea of analogous events, or, more generally, anal-ogous random elements into probabilistic terms. Inthe last analysis, such an assumption stems from asubjective judgment, and the solution that the prob-lem of induction finds through the asymptotic theo-rem for predictive distributions (Section 2.3) is ob-viously subjective:

but in itself perfectly logical, while onthe other hand, when one pretends toeliminate the subjective factors one suc-ceeds only in hiding them : : :more orless skilfully, but never in avoiding agap in logic. It is true that in manycases—as for example in the hypothe-sis of exchangeability—these subjectivefactors never have too pronounced an in-fluence, provided that the experience berich enough; this circumstance is veryimportant, for it explains how in certainconditions more or less close agreementbetween the predictions of different in-dividuals is produced, but it also showsthat discordant opinions are always le-gitimate.” [From the English translationof La prevision in Kyburg and Smokler,1980, page 107.]

A second circumstance which distinguishesde Finetti’s position from the usual Bayesianstatement of statistical inference is his caution to-wards the usual interpretation in terms of “randomelements independent and identically distributedconditional on a random parameter.” In order toexplain this view, de Finetti resorts to two exam-ples, at once quite simple and yet representative

of situations of doubtless practical meaning: thesequence of drawings with replacement from anurn containing both white and nonwhite balls ac-cording to an unknown composition; the sequenceof tosses of a same coin. The sequence of events�En�n≥1 (with En the drawing of a white ball inthe nth drawing, En is heads in the nth toss) canbe viewed as exchangeable in both cases. Conse-quently, the representation theorem formulated inSection 2.3 holds that from a formal viewpoint thelaw of the sequence is just a mixture of Bernoullianprocesses. Again, from a merely formal viewpoint,exchangeable events correspond to those that arecommonly viewed as independent with constant butunknown probability θ, where θ is distributed ac-cording to the mixing distribution which appearsin de Finetti’s representation theorem. Why thenstart out on a new definition and go to great lengthsjust to prove something well known, we may won-der. Here is the answer. The old definition cannotpossibly be stripped of its metaphysical character.In its light, you are bound to admit that, beyondthe probability law corresponding to your subjec-tive judgment (the exchangeable law) there must“exist” another law, quite unknown, which corre-sponds to something real. The diverse hypotheseswhich might then be made on this unknown law,according to which events would no longer be de-pendent, but actually stochastically independent,would in turn constitute events whose probabilityshould then be duly evaluated. To de Finetti’s wayof thinking, that is to say within the framework ofhis subjectivistic approach, all this does not actuallymake any sense. In the case of the urn of unknowncomposition, we could no doubt speak of the proba-bility of the composition of the urn as well as of theconditional probability given each composition. Inpoint of fact, the composition is empirically verifi-able. On the other hand, if you play heads or tailsand toss an “unfair” coin, you cannot infer that thisvery unfairness has a given influence over the “un-known probability” and hence view this unfairnessas an observable hypothesis. In fact, this “unknownprobability” cannot possibly be defined, and all thehypotheses one would like to introduce in this wayhave no objective meaning.

The difference between the two previous casesis quite substantial and cannot be neglected. Onecannot “by analogy” recover in the second case thereasoning which was valid for the urn, for thisreasoning no longer applies in the second case. Inother words, it is plainly impossible to stick to theusual fuzzy interpretation. Instead, if one wantsto go about it in a rigorous way, one should tackleboth cases by consistently starting from what they


do have in common, that is, the similarity of theenvironmental conditions in which both the suc-cessive drawings and the successive tosses takeplace. This is the very same similarity which is con-ducive to the adoption of the probabilistic conditionof exchangeability. This way it is exchangeabilitywhich presides over inductive reasoning in the pres-ence of the classical conditions which characterizethe statistical version of such a kind of reason-ing. Moreover, as we have seen in Section 2.3, allreasonings regarding events must be extended torandom elements taking values in arbitrary spaces.

The message from these thought-provoking re-marks is clear enough: statistical inference mustconfine itself to considering objective hypotheses onsomething that can be really observed, at least inprinciple. Hence, probabilistic models, introduced toformalize statistical problems ought first and fore-most to consider finitary events but they ought notto fix compulsory rules in connection with the ex-tension of probabilities from finitary to infinitaryevents. In fact, while it is reasonable to believethat one is able to represent, in some acceptableway, one’s judgment of probability on observablefacts, it is illusory to think that one might sensi-bly express any judgment about something whichhas no empirical meaning, like infinitary events.Hence, any attempt to lend objectivity or any em-pirical interpretation to entities which originatefrom any extension of a probability to infinitaryevents (through purely mathematical arguments) isgenerally in conflict with the difficulty of justifyinga special status (from a logical standpoint) for anyextension of the type quoted above. Diaconis (1988)is very clear on this point and he deserves to becited directly:

de Finetti’s alarm at statisticians intro-ducing reams of unobservable parame-ters has been repeatedly justified in themodern curve fitting exercises of today’sbig models. These seem to lose all con-tact with scientific reality focusing atten-tion on details of large programs and fit-ting instead of observation and under-standing of basic mechanism. It is to behoped that a fresh implementation of deFinetti’s program based on observableswill lead us out of this mess.

For these reasons, de Finetti’s representation the-orem, in view of statistical interpretations, ought tobe enounced according to its original “weak” formu-lation (Section 2.3) or, in other words, one oughtto avoid the seemingly more enticing strong for-mulation “the observable Xn’s are exchangeable if

and only if they, conditional on a random elementθ, are independent and identically distributed”. In-deed, the strong formulation could lead to justifyinginference on θ even when θ is not observable. Tostrengthen this position, let us recall that the exis-tence of θ, in general, is a mere mathematical state-ment, due to some particular extension of a fam-ily of coherent finite-dimensional distributions to acomplete law for the entire sequence �Xn�n≥1. Ex-ample 3.1 in Regazzini and Petris (1992) shows thatthere are sequences of exchangeable events whosecorresponding frequency of success does not con-verge in any well-specified stochastic sense. Theseremarks lead us to focus on de Finetti’s insistence onthe importance of the weak approach (based on fi-nite additivity) to probability. In general, one thinksthat the dilemma “finite versus complete additivity”concerns mathematics or philosophy, so that thereare “subjectivistic Bayesian statisticians” who justthink of the issue as simply trivial. In spite of this,the previous remarks show that the very meaningof statistical inference may depend on the solutionenvisaged.

Up to now we have dealt with statistics as induc-tive reasoning on analogous observable random ele-ments, which justifies the assumption of exchange-ability when it comes to those random elements.In this case, induction is independent of the orderof observations. Apropos of this, de Finetti, in Laprevision, points out:

One can indeed take account not only ofthe observed frequency, but also of regu-larities or tendencies toward certain reg-ularities which the observations can re-veal. Suppose, for example, that the nfirst trials give alternately a favourableresult and an unfavourable result. In thecase of exchangeability, our prediction ofthe following trial will be the same afterthese n trials as after any other experi-ence of the same frequency of 1/2, butwith a completely irregular sequence ofdifferent results; it is indeed the absenceof any influence of the order on the judg-ments of a certain individual which char-acterizes, by definition, the events thathe will consider “exchangeable”. In thecase where the events are not conceivedof as exchangeable, we will, on the otherhand, be led to modify our predictionsin a very different way after n trials ofalternating results than after n irregu-larly disposed trials having the same fre-quency of 1/2; the most natural attitude


will consist in predicting that the nexttrial will have a great probability of pre-senting a result opposite to that of thepreceding trial.

It would doubtless be possible andinteresting to study this influence oforder in some simple hypotheses, bysome more or less extensive generaliza-tion of the case of exchangeability, andsome developments tied up with thatgeneralization : : : : [From the Englishtranslation in Kyburg and Smokler,1980, page 106.]

At that time, a comprehensive survey of the issuewas still to be carried out. A first generalization tobe mentioned is the definition of partial exchange-ability by de Finetti himself; see Section 2.3. In Laprobabilita e la statistica, he emphasizes the role ofthis generalization in inductive reasoning and givesa good piece of advice on how to fit Markov depen-dence into partial exchangeability.

Note. The trail blazed by de Finetti and lead-ing to reconsidering the Bayes–Laplace paradigmhas been followed by a small group of scholars, asshown in Section 2.3; see also Diaconis (1988). Withreference to the Italian part of this group, first andforemost let us mention the book by Daboni andWedlin (1982), who reformulate the basic statisticalresults according to de Finetti’s approach. OtherItalian scholars have considered characterizationsof statistical models starting from exchangeabilitycondition combined with a new notion of predictivesufficiency; see Campanino and Spizzichino (1981),Cifarelli and Regazzini (1982) and Spizzichino(1988). In any case, the last-mentioned subjectis connected with the previously cited works byDiaconis and Freedman.

3.2.2 de Finetti, the objectivistic schools and thetheory of decisions. De Finetti’s attitude towardthe so-called objectivistic schools was extremelycritical, although he thought of the rise of objec-tivistic conceptions as a reaction to unacceptableand discredited formulations of the Bayes–Laplaceparadigm, based on mysterious “unknown probabil-ities.” In his opinion:

Fisher’s rich and manifold personalityshows a few contradictions. His com-mon sense in applications on one handand his lofty conception of scientific re-search on the other lead him to disdainthe narrowness of a genuinely objec-tivistic formulation, which he regarded

as a “wooden attitude.” He professeshis adherence to the objectivistic pointof view by rejecting the errors of theBayes–Laplace formulation. What is notso good here is his mathematics, whichhe handles with mastery in individualproblems but rather cavalierly in con-ceptual matters, thus exposing himselfto clear and sometimes heavy criticism.From our point of view it appears prob-able that many of Fisher’s observationsand ideas are valid provided we go backto the intuitions from which they springand free them from the arguments bywhich he thought to justify them. On theother hand, the coherent developmentof a theory of statistical induction onrigidly objectivistic bases is the specificcharacteristic of the Neyman–Pearsonschool. There, probability means noth-ing but “frequency in the long run,” andthis also includes moments of distractionfor the sake of convenience. To accepteither a hypothesis or an estimate doesnot mean to attribute to it any kind ofprobability or plausibility. Such accep-tance is a mechanical act based not ona judgment of its actual validity buton the frequency of the validity of themethod from which it is derived”. [Fromthe English translation of La probabilitae la statistica in de Finetti, 1972, pages171–172.]

De Finetti met Neyman at the Geneva Confer-ence in 1937. On that occasion, Neyman delivereda lecture on statistical estimation. De Finetti’s com-ment on Neyman’s lecture and Neyman’s answer tode Finetti’s comment illustrate their disagreementon the role of Bayes’ theorem; see de Finetti (1939a).

Insofar as the theory of statistical decision isconcerned, de Finetti interpreted the advent ofthe Wald decision theoretical approach to statis-tical inference as “the most decisive involuntarycontribution toward the erosion of the objectivisticpositions,” in connection with the basic statementthat admissible decisions are Bayesian. In fact,he viewed that theory as a modern developmentof Bernoulli’s concept of moral expectation, quitesuitable for furthering a reconciliation betweendifferent viewpoints inspired, on one hand, byde Finetti’s reconstruction of the Bayes–Laplaceparadigm and, on the other, by the neo-Bernoullianview of statistical problems. Insofar as we know, hedid not obtain new results in decision theory, but


he did produce an interesting interpretation of theBernoulli principle of the moral expectation, basedon the de Finetti–Kolmogorov–Nagumo theorem forassociative means (Section 3.1). Indeed, by assum-ing that all random gains take values in �A;B�,we can identify the class of all these gains withF�A;B�. Then, if one assumes that a person P isable to associate a real number m�F� (called thevalue of F) with each F in F�A;B�, in such a waythat it is indifferent for P to possess F or m�F�,then m: F�A;B� → R is a mean. If, in comparingelements of F�A;B�, P clings to the following rules:

(a) F = Dα ⇒ m�F� = α;(b) m is strictly monotone and associative (for the

economic meaning of these concepts see Section 3.1);

then, from the aforementioned theorem, one has

m�F� = φ−1(∫�A;B�

φ�x�dF�x�);

where φ is continuous and strictly increasing. Thisfunction—the utility function of money—can bedetermined by iterating the following experimentwhere, without loss of generality, it is assumed thatφ�A� = A and φ�B� = B. Hence P , in order to de-termine φ on �A;B�, can assess the value g1 of arandom gain which takes the values A and B withprobability 1/2 and 1/2, respectively. Then

g1 = φ−1 ( 12φ�A� + 1

2φ�B�);

which yields

φ�g1� = 12�A+B�:

Iterating this process, P will assess the value g2of a random gain which takes the values g1 and Bwith probability 1/2 and 1/2, respectively. In thisway, one determines φ�g2� and so on.

The function

F→∫�A;B�

φdF

is known as preferability index or expected utility. Ina sense, the representation theorem for associativemeans, according to de Finetti’s approach, allows usto reach the main conclusion of the von Neumann–Morgenstern theory. De Finetti’s line of reasoningabout the theory of statistical decisions, as opposedto that of other scholars holding different views, isclearly expounded in an interesting survey by Pic-cinato (1986), who is particularly attentive to alldevelopments in the field. De Finetti was also inter-ested in group decision-making. His results in thisarea have recently been taken up by Barlow, Wech-sler and Spizzichino (1988), who suggest an inter-esting generalization on the matter.

With reference to decision-making, we must notforget that it was in this field of investigation thatde Finetti encountered Leonard J. Savage. Theirrelationship was mutually stimulating. Savageguided de Finetti through objectivistic methods andhelped him gain access to wider statistical circles;de Finetti urged Savage to go deeper into the newand exciting perspectives by the neo-Bayesian andneo-Bernoullian thought.

When expounding his thought, de Finetti was al-ways uncompromising. He actually never spared ac-curate criticism against the views he did not share.Yet he was also eager to find some common groundfor all statistical views. The title of one of his pa-pers, “Recent suggestions for the reconciliation oftheories of probability” (de Finetti, 1951), is enlight-ening. Let us quote from it:

From the subjective standpoint, no as-sertion is possible without a priori opin-ion, but the variety of possible opinionsmakes problems depending on differentopinions interesting.

And this sounds, indeed, as a wholehearted wel-come to our conference on Bayesian robustness.

APPENDIX 1:GENEVA CONFERENCE LECTURES,

OCTOBER 1937

• Frechet (Paris): On some recent advances in thetheory of probability;• Steffensen (Copenhagen): Frequency and prob-

ability;• Wald (Wien): Compatibility of the notion of

“Kollektiv”;• Feller (Stockholm): On the foundations of prob-

ability;• Neyman (London): Probabilistic treatment of

some problems in mathematical statistics;• de Finetti (Trieste): On the condition of partial

exchangeability;• Heisenberg (Leipzig): Probabilistic statements

in the wave quantum theory;• Hopf (Leipzig): Statement of mechanics prob-

lems according to measure theory;• Polya (Zurich): Random walk in a road network;• Hostinsky (Brno): Random change–fluctuations

of the number of objects in a compartment;• Onicescu (Bucharest): Sketch of a general theory

for chains with complete bounds;• Dodd (Austin): Certain coefficients of regression

or trend associated with largest likelihood;• Cramer (Stockholm): On a new central limit the-

orem;


• Obrechkoff (Sofiya): Difference equations withconstant coefficients, the Poisson distribution and theCharlier series;• Steinhaus (Lvov): Theory and applications of

stochastically independent functions;• Levy (Paris): Arithmetic of probability distribu-

tions.

As a matter of fact, other thought-provoking con-tributions were on the agenda, namely:

• Bernstein (Leningrad): Theory of stochastic dif-ferential equations;• Cantelli (Rome): About the definition of random

variable;• Glivenko (Moscow): About the strong law of

large numbers in a functions space;• Jordan (Budapest): Critical aspects of the corre-

lation theory from a probabilistic viewpoint;• Kolmogorov (Moscow): Random functions and

their applications;• von Mises (Istanbul): Generalization of classical

limit theorems;• Romanowsky (Tashkent): Some new results in

the theory of Markov chains;• Slutsky (Moscow): Some statements in the the-

ory of random functions with a continuous spectrum.

Unfortunately these last-mentioned lecturers didnot turn up. However, the complete texts or atleast summaries of the lectures were publishedin Actualites Scientifiques et Industrielles, includ-ing a slightly modified French translation of deFinetti’s Resoconto Critico del Colloquio; see deFinetti (1939a).

ACKNOWLEDGMENTS

We are grateful to Gianfranco Barbieri, IleniaEpifani and Antonio Lijoi for having helped us,in various ways, with the draft of the present pa-per. Extremely useful comments were providedby two anonymous referees. This work was par-tially supported by the MURST (40%, 1994, “In-ferenza Statistica: basi probabilistiche e sviluppimetodologici”) and by Universita Bocconi (Ricercadi base 1994–1995).

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Cantelli, F. P. (1932). Una teoria astratta del calcolo delle pro-babilita. Giorn. Istit. Ital. Attuari 3 257–265.

Cantelli, F. P. (1933a). Considerazioni sulla legge uniforme deigrandi numeri e sulla generalizzazione di un fondamentaleteorema del signor Levy. Giorn. Istit. Ital. Attuari 4 327–350.

Cantelli, F. P. (1933b). Sulla determinazione empirica delleleggi di probabilita. Giorn. Istit. Ital. Attuari 4 421–424.

Castellano, V. (1965). Corrado Gini: a memoir. Metron 24 3–84.Castelnuovo, G. (1919). Calcolo delle Probabilita e Appli-

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Cifarelli, D. M. and Regazzini, E. (1987). On a general defini-tion of concentration function. Sankhya Ser. B 49 307–319.

Cifarelli, D. M. and Regazzini, E. (1990). Distribution func-tions of means of a Dirichlet process. Ann. Statist. 18 429–442. [Correction: Ann. Statist. (1994) 22 1633–1634.]

Cifarelli, D. M. and Regazzini, E. (1993). Some remarks onthe distribution functions of means of a Dirichlet process.Technical Report 93.4, CNR-IAMI, Milano.

Cramer, H. (1934). Su un teorema relativo alla legge uniformedei grandi numeri. Giorn. Istit. Ital. Attuari 5 3–15.

Cramer, H. (1935). Sugli sviluppi asintotici di funzioni di ripar-tizione in serie di polinomi di Hermite. Giorn. Istit. Ital. At-tuari 6 141–157.

Crisma, L. (1971). Alcune valutazioni quantitative interessantila proseguibilita di processi aleatori scambiabili. Rend. Istit.Mat. Univ. Trieste 3 96–124.

Crisma, L. (1982). Quantitative analysis of exchangeability inalternative processes. In Exchangeability in Probability andStatistics (G. Koch and F. Spizzichino, eds.) 207–216. North-Holland, Amsterdam.

Daboni, L. (1953). Considerazioni geometriche sulla condizionedi equivalenza per una classe di eventi. Giorn. Istit. Ital.Attuari 16 58–65.

Daboni, L. (1975). Caratterizzazione delle successioni (funzioni)completamente monotone in termini di rappresentabilitadelle funzioni di sopravvivenza di particolari intervalli scam-biabili tra successi (arrivi) contigui. Rend. Mat. 8 399–412.

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Diaconis, P. and Freedman, D. (1980a). de Finetti’s theorem forMarkov chains. Ann. Probab. 8 115–130.

Diaconis, P. and Freedman, D. (1980b). Finite exchangeable se-quences. Ann. Probab. 8 745–764.

Diaconis, P. and Freedman, D. (1984). Partial exchangeabilityand sufficiency. In Proceedings of the Indian Statistical In-stitute Golden Jubilee International Conference on Statistics:Applications and New Directions (J. K. Ghosh and J. Roy,eds.) 205–236. Indian Statistical Institute, Calcutta.

Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-styleresults in search of a theory. Ann. Inst. H. Poincare Probab.Statist. 23 394–423.

Diaconis, P. and Freedman, D. (1988). Conditional limit the-orems for exponential families and finite versions of deFinetti’s theorem. J. Theoret. Probab. 1 381–410.

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Dubins, L. E. (1975). Finitely additive conditional probabilities,conglomerability and disintegrations. Ann. Probab. 3 89–99.

Dubins, L. E. (1983). Some exchangeable probabilities whichare singular with respect to all presentable probabilities.Z. Wahrsch. Verw. Gebiete 64 1–6.

Dubins, L. E. and Freedman, D. (1979). Exchangeable processesneed not be mixtures of independent identically distributedrandom variables. Z. Wahrsch. Verw. Gebiete 48 115–132.

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Epifani, I. and Lijoi, A. (1995). Some considerations on a versionof the law of the iterated logarithm due to Cantelli. TechnicalReport 95.21, CNR-IAMI, Milano.

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Fichera, G. (1987). Interventi invitati al Convegno “Ricordodi Bruno de Finetti Professore nell’Ateneo triestino.” Atti26–31. Dip. Matematica Applicata alle Scienze EconomicheStatistiche e Attuariali, Univ. Trieste Pub. No. 1.

Fisher, R. A. (1921). On “probable error” of a coefficient of cor-relation deduced from a small sample. Metron 1 3–32.

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Fortini, S., Ladelli, L. and Regazzini, E. (1996). A central limitproblem for partially exchangeable random variables. TheoryProbab. Appl. 41 353–379.

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Frechet, M. (1930c). Sur la convergence “en probabilite.” Metron8 3–50.

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Glivenko, V. (1933). Sulla determinazione empirica delle leggidi probabilita. Giorn. Istit. Ital. Attuari 4 92–99.


Glivenko, V. (1936). Sul teorema limite della teoria delle fun-zioni caratteristiche. Giorn. Istit. Ital. Attuari 7 160–167.

Glivenko, V. (1938). Theorie Generale des Structures. Hermann,Paris.

Haag, J. (1928). Sur un probleme general de probabilites etses diverses applications. In Proceedings of the InternationalCongress of Mathematicians 659–674. Univ. Toronto Press.

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Kallenberg, O. (1982). Characterizations and embedding prop-erties in exchangeability. Z. Wahrsch. Verw. Gebiete 60 249–281.

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Kolmogorov, A. N. (1932b). Ancora sulla forma generale di unprocesso omogeneo. Atti R. Acc. Lincei, Serie VI, Rendiconti15 866–869.

Kolmogorov, A. N. (1933a). Sulla determinazione empirica diuna legge di distribuzione. Giorn. Istit. Ital. Attuari 4 92–99.

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Kuchler, V. and Lauritzen, S. L. (1989). Exponential familiesextreme point models, and minimal space–time invariantfunctions for stochastic processes with stationary and inde-pendent increments. Scand. J. Statist. 15 237–261.

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Levy, P. (1931a). Sulla legge forte dei grandi numeri. Giorn. Is-tit. Ital. Attuari 2 1–21.

Levy, P. (1931b). Sur quelques questions de calcul des proba-bilites. Prace Matematyezno Fizyczne �Warsaw� 39 19–28.

Levy, P. (1934). Sur les integrales dont les elements sont desvariables aleatoires independantes. Ann. Scuola Norm. Sup.Pisa 3 337–366; 4 217–235.

Levy, P. (1935). Sull’applicazione della geometria dello spaziodi Hilbert allo studio delle successioni di variabili casuali.Giorn. Istit. Ital. Attuari 6 13–28.

Lundberg, F. (1903). Approximerad framstallning av sanno-likhetsfunktionen. Återforsakring av kollektivrisker. Thesis,Uppsala.

Nagumo, M. (1930). Uber eine Klasse von Mittelwenten. JapanJ. Math. 7 71–79.

Neyman, J. (1935). Su un teorema concernente le cosiddettestatistiche sufficienti. Giorn. Istit. Ital. Attuari 6 320–334.

Ottaviani, G. (1939). Sulla teoria astratta del calcolo delle pro-babilita proposta dal Cantelli. Giorn. Istit. Ital. Attuari 1010–40.

Paley, R. E., Wiener, N. and Zygmund, A. (1933). Note on ran-dom functions. Math. Z. 37 647–668.

Petris, G. and Regazzini, E. (1994). About the characterizationof mixtures of Markov chains. Unpublished manuscript.

Piccinato, L. (1986). De Finetti’s logic of uncertainty and itsimpact on statistical thinking and practise. In Bayesian In-ference and Decision Techniques (P. K. Goel and A. Zellner,eds.) 13–30. North-Holland, Amsterdam.

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Regazzini, E. and Petris, G. (1992). Some critical aspects of theuse of exchangeability in statistics. Journal of the ItalianStatistical Society 1 103–130.

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Romanowsky, V. (1925). On the moments of standard deviationand of correlation coefficient in samples from normal. Metron5 3–46.

Romanowsky, V. (1928). On the criteria that two given samplesbelong to the same normal population. Metron 7 3–46.

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Slutsky, E. (1934). Alcune applicazioni dei coefficienti di Fourierall’analisi delle funzioni aleatorie stazionarie. Giorn. Istit.Ital. Attuari 5 435–482.

Slutsky, E. (1937). Alcune proposizioni sulla teoria delle fun-zioni aleatorie. Giorn. Istit. Ital. Attuari 8 183–199.

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Spizzichino, F. (1982). Extendibility of symmetric probabilitydistributions and related bounds. In Exchangeability inProbability and Statistics (G. Koch and F. Spizzichino, eds.)313–320. North-Holland, Amsterdam.

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GENERAL REFERENCES CONCERNINGDE FINETTI’S WORK

(1987). Atti del Convegno “Ricordo di Bruno de Finetti Professorenell’Ateneo triestino.” Dip. Matematica applicata alle scienzeeconomiche statistiche e attuariali. “Bruno de Finetti” Pub.No. 1.

Daboni, L. (1987). Bruno de Finetti. Boll. Un. Mat. Ital. A 283–308.

Gani, J., ed. (1982). The Making of Statisticians. Springer, NewYork.

Goel, P. K. and Zellner, A., eds. (1986). Bayesian Inference andDecision Techniques. Essays in Honor of Bruno de Finetti.North-Holland, Amsterdam.

Koch, G. and Spizzichino, F., eds. (1982). Exchangeability inProbability and Statistics. North-Holland, Amsterdam.

Kotz, S. and Johnson, N. L., eds. (1992). Breakthroughsin Statistics 1. Foundations and Basic Theory. Springer,New York.

Kyburg, H. E., Jr. and Smokler, H. E., eds. (1980). Studies inSubjective Probability. Krieger, Malabar, FL.

(1987). La Matematica Italiana tra le Due Guerre Mondiali. Mi-lano, Gargnano del Garda, 8–11 ottobre 1986. Pitagora Ed.,Bologna.

Lindley, D. V. (1989). De Finetti, Bruno. In Encyclopedia of Sta-tistical Sciences (Supplement) 46–47. Wiley, New York.

Regazzini, E. (1987). Probability theory in Italy between the twoworld wars. A brief historical review. Metron 44 5–42.

von Plato, J. (1994). Creating Modern Probability. CambridgeUniv. Press.

CITED WORKS OF DE FINETTI

Articles

(1929a). Sulle funzioni a incremento aleatorio. Atti. Reale Ac-cademia Nazionale dei Lincei, Serie VI, Rend. 10 163–168.

(1929b). Sulla possibilita di valori eccezionali per una legge adincrementi aleatori. Atti Reale Accademia Nazionale dei Lin-cei, Serie VI, Rend. 10 325–329.

(1929c). Integrazione delle funzioni a incremento aleatorio. AttiReale Accademia Nazionale dei Lincei, Serie VI, Rend. 10548–552.

(1930a). Sui passaggi al limite nel calcolo delle probabilita. Ren-diconti del Reale Istituto Lombardo di Scienze e Lettere 63155–166.

(1930b). A proposito dell’estensione del teorema delle probabilitatotali alle classi numerabili. Rendiconti del Reale IstitutoLombardo di Scienze e Lettere 63 901–905.

(1930c). Ancora sull’estensione alle classi numerabili del teo-rema delle probabilita totali. Rendiconti Reale Istituto Lom-bardo di Scienze e Lettere 63 1063–1069.

(1930d). Problemi determinati e indeterminati nel calcolo delleprobabilita. Atti Reale Accademia Nazionale dei Lincei, SerieVI, Rend. 12 367–373.

(1930e). Sulla proprieta conglomerativa delle probabilita subor-dinate. Atti Reale Accademia Nazionale dei Lincei, Serie VI,Rend. 12 278–282.

(1930f ). Introduzione matematica alla statistica metodologica.Riassunto di un corso di lezioni per laureati, Istituto Cen-trale di Statistica, Roma.

(1930g). Funzione caratteristica di un fenomeno aleatorio. AttiReale Accademia Nazionale dei Lincei, Mem. 4 86–133.

(1930h). Le funzioni caratteristiche di legge istantanea. AttiReale Accademia Nazionale dei Lincei, Serie VI, Rend. 12278–282.

(1930i). Calcolo della differenza media. Metron 8 89–94 (withU. Paciello).

(1931a). Probabilismo. Saggio critico sulla teoria della prob-abilita e sul valore della scienza. Logos �Biblioteca diFilosofia� 163–219. Perrella, Napoli.

(1931b). Sul significato soggettivo della probabilita. FundamentaMathematicae 17 298–329.

(1931c). Le funzioni caratteristiche di legge istantanea dotate divalori eccezionali. Atti Reale Accademia Nazionale dei Lincei,Serie VI, Rend. 14 259–265.

(1931d). Sui metodi proposti per il calcolo della differenza media.Metron 9 47–52.

(1931e). Sul concetto di media. Giorn. Istit. Ital. Attuari 2 369–396.

(1931f ). Calcoli sullo sviluppo futuro della popolazione italiana.Annuali di Statistica 10 3–130 (with C. Gini).

(1931g). Le leggi differenziali e la rinunzia al determinismo. Ren-diconti del Seminario Matematico dell’Universita di Roma 763–74.

(1932a). Funzione caratteristica di un fenomeno aleatorio. InAtti del Congresso Internazionale dei Matematici 179–190.Zanichelli, Bologna.


(1932b). Sulla legge di probabilita degli estremi. Metron 9 127–138.

(1933a). Classi di numeri aleatori equivalenti. La legge deigrandi numeri nel caso dei numeri aleatori equivalenti. Sullalegge di distribuzione dei valori in una successione di numerialeatori equivalenti (Three articles). Atti Reale AccademiaNazionale dei Lincei, Serie VI, Rendiconti 18 107–110; 203–207; 279–284.

(1933b). Sull’approssimazione empirica di una legge di proba-bilita. Giorn. Istit. Ital. Attuari 4 415–420.

(1935a). Sull’integrale di Stieltjes–Riemann. Giorn. Istit. Ital.Attuari 6 303-319 (with M. Jacob).

(1935b). Il problema della perequazione. In Atti della SocietaItaliana per il Progresso delle Scienze (XXIII Riunione,Napoli, 1934) 2 227–228. SIPS, Roma.

(1937a). La prevision: ses lois logiques, ses sources subjectives.Ann. Inst. H. Poincare 7 1–68. [English translation in Studiesin Subjective Probability (1980) (H. E. Kyburg and H. E.Smokler, eds.) 53–118. Krieger, Malabar, FL.]

(1937b). A proposito di “correlazione.” Supplemento Statistico aiNuovi Problemi di Politica, Storia ed Economia 3 41–57.

(1938a). Sur la condition de “equivalence partielle.” In ActualitesScientifiques et Industrielles 739 5–18. Hermann, Paris.

(1938b). Funzioni aleatorie. In Atti del I Congresso UMI 413–416. Zanichelli, Bologna.

(1938c). Resoconto critico del colloquio di Ginevra intorno allateoria delle probabilita. Giorn. Istit. Ital. Attuari 9 1–40.

(1939a). Compte rendu critique du colloque de Geneve sur latheorie des probabilites. In Actualites Scientifiques et Indus-trielles 766. Hermann, Paris.

(1939b). La teoria del rischio e il problema della “rovina dei gio-catori.” Giorn. Istit. Ital. Attuari 10 41–51.

(1939c). Indici statistici e “teoria delle strutture.” In Supple-mento Statistico ai Nuovi Problemi di Politica, Storia edEconomia 5 71–74. S.A.T.E., Ferrara.

(1943). La matematica nelle concezioni e nelle applicazionistatistiche. Statistica 3 89–112.

(1949). Sull’impostazione assiomatica del calcolo delle proba-bilita. Aggiunta alla nota sull’assiomatica della probabilita(Two articles). Annali Triestini 19 sez. II 29–81; 20 sez. II3–20.

(1951). Recent suggestions for the reconciliation of theories

of probability. Proc. Second Berkeley Symp. Math. Statist.Probab. 217–225. Univ. California Press, Berkeley.

(1952). Gli eventi equivalenti e il caso degenere. Giorn. Istit. Ital.Attuari 15 40–64.

(1953a). Trasformazioni di numeri aleatori atte a far coincideredistribuzioni diverse. Giorn. Istit. Ital. Attuari 16 51–57.

(1953b). Sulla nozione di “dispersione” per distribuzioni a piu di-mensioni. In Atti del IV Congresso UMI 587–596. Cremonese,Roma.

(1955a). La struttura delle distribuzioni in un insieme astrattoqualsiasi. Giorn. Istit. Ital. Attuari 18 12–28. [English trans-lation in B. de Finetti, Probability, Induction and Statistics(1972) 129–140. Wiley, New York).]

(1955b). Sulla teoria astratta della misura e dell’integrazione.Ann. Mat. Pura Appl. 40 307–319. [English translation inB. de Finetti, Probability, Induction and Statistics (1972)115–128. Wiley, New York.]

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