Topics in Stochastic Geometry

Convex Hullls of Random Walks

Lecture Notes for GR8260, Spring 2020, Columbia University

Julien Randon-Furling

Classes: Tue 11:40am & Thu 2:40pm, Mathematics Hall Room 622

Office hours (Room 625): Mon to Fri 9am - 2pm or appointment

Abstract This course covers a range of results on the convex hull of random walks in the planeand in higher dimension: expected perimeter length in the planar case, expected number offaces on the boundary, expected d-dimensional volume, and other geometric properties of suchrandom convex polytopes.Beforehand, we will review a number of classical one-dimensional results such as the Arc-sine laws. Moving on to higher-dimensional random walks, we will introduce the fundamentalobject of this course, that is the joint convex hull of a collection of m random walks:

Cd = conv(

0, S(1)1 , . . . , S(1)

n1, . . . , S

(m)1 , . . . , S(m)

nm

),

with (S(1)i )n1

i=1, . . . , (S(m)i )nmi=1 defined as

S(l)i = X

(l)1 + · · ·+X

(l)i , 1 ≤ l ≤ m, 1 ≤ i ≤ nl,

where m,n1, . . . , nm ∈ N, and

X(1)1 , . . . , X(1)

n1, . . . , X

(m)1 , . . . , X(m)

nm

are independent d-dimensional random vectors drawn from the same distribution (eg Gaussian).

The case m = 1 is the d-dimensional random walk; the case m ≥ d+ 1 and ∀l ≥ 1, nl = 1 is,when the underlying distribution is the Gaussian one, the standard Gaussian polytope. Othercases include multiple, independent d-dimensional random walks.In dimension 2 the course will cover results such as the Spitzer-Widom formula for the expectedcircumference of the convex hull, and Baxter’s combinatorial lemma for the expected numberof edges on the boundary of the convex hull. In higher dimension, we will examine recentresults on the expected number of faces of the boundary of the convex hull, including the caseof multiple random walks.In the last part of the course, we will discuss related topics such as the convex hull of Brownian

1

motion (in the plane and in higher dimensions) together with connections to one-dimensionalresults on the greatest convex minorant of random walks, Brownian motion and more generalLevy processes. While doing so, we will review classical results on excursions, meanders andbridges.

PrerequisiteBasic knowledge of probability theory, random walks, stochastic processes.

Suggested reading

• Andersen, E. S. (1949). On the number of positive sums of random variables. SkandinaviskAktuarietidskrift, 32, 27-36.

• Andersen, E. S. (1954). On the fluctuations of sums of random variables I & II. MathematicaScandinavica, 263-285 & (1955) 195-223.

• Spitzer, F., & Widom, H. (1961). The Circumference of a Convex Polygon. Proceedings of theAmerican Mathematical Society, 12(3), 506-509.

• Baxter, G. (1961). A combinatorial lemma for complex numbers. The Annals of MathematicalStatistics, 32(3), 901-904.

• Barndorff-Nielsen, O., & Baxter, G. (1963). Combinatorial lemmas in higher dimensions. Trans-actions of the American Mathematical Society, 108(2), 313-325.

• Takacs, L. (1980). Problems and Solutions: Solutions of Advanced Problems: 6230. Amer.Math. Monthly, 87(2), 142.

• Evans, S. N. (1985). On the Hausdorff dimension of Brownian cone points. MathematicalProceedings of the Cambridge Philosophical Society (Vol. 98, No. 2, pp. 343-353). CambridgeUniversity Press.

• Snyder, T., & Steele, J. (1993). Convex Hulls of Random Walks. Proceedings of the AmericanMathematical Society, 117(4), 1165-1173.

• Majumdar, S. N., Comtet, A., & Randon-Furling, J. (2010). Random convex hulls and extremevalue statistics. Journal of Statistical Physics, 138(6), 955-1009.

• Kampf, J., Last, G., & Molchanov, I. (2012). On the convex hull of symmetric stable processes.Proceedings of the American Mathematical Society, 140(7), 2527-2535.

• Biane, P. & Letac, G. (2011). The Mean Perimeter of Some Random Plane Convex Sets Gen-erated by a Brownian Motion. Journal of Theoretical Probability, 24: 330.

• Eldan, R. (2014). Volumetric properties of the convex hull of an n-dimensional Brownian motion.Electronic Journal of Probability, 19.

• Kabluchko, Z., Vysotsky, V., & Zaporozhets, D. (2017). Convex hulls of random walks: Ex-pected number of faces and face probabilities. Advances in Mathematics, 320, 595-629.

• Vysotsky, V., & Zaporozhets, D. (2018). Convex hulls of multidimensional random walks.Transactions of the American Mathematical Society, 370(11), 7985-8012.

• Randon-Furling, J. & Zaporozhets, D. (2019/2020, to appear). Convex hull of multiple multi-dimensional Gaussian random walks. Preprint

• Pitman, J. & Ross, N. (2012). The greatest convex minorant of Brownian motion, meander,and bridge. Probability Theory and Related Fields, 153(3-4), 771-807.

• Pitman, J. & Uribe Bravo, G. (2012). The convex minorant of a Levy process. The Annals ofProbability, 40(4), 1636-1674.

• Chung, K.L. (1976). Excursions in Brownian motion. Arkiv for Matematik, 14(1-2), 155-177.

2

1 Positive points on the path of a random walk

We start by reviewing a fundamental theorem pertaining to the number of positive points onthe path of a one-dimensional random walk. Namely, if X1, X2, . . . are real random variables(at this stage we do not make any assumption on their joint distribution), one defines thecorresponding random walk as

Sn = X1 +X2 + · · ·+Xn,

and we set S0 = 0.It is customary to call X1, X2, . . . the “jumps” of the random walk, and S1, S2, S3, . . . , Sn the“positions” or simply the “points” of the random walk. The set {(k, Sk), 0 ≤ k n} is oftenreferred to as the “path” of the random walk, that is, its set of “steps”. If we prefer viewinga random walk simply as a sum of random variables, then S0, S1, S2, S3, . . . , Sn is just thesequence of partial sums.Now define the following random variable:

Nn =n∑k=0

1{Sk>0},

i.e. the number of positive points in the first n steps of the random walk (1 denotes an indicatorfunction). The law of Nn was computed by Erik Sparre Andersen in 1949. Before we state hisresult, we need the following definition.

Definition 1. An s-permutation (x1, . . . , xn) of n real numbers (a1, . . . , an) is a permutation(εi1ai1 , . . . , εinain) of the n numbers (ε1a1, . . . , εnan) where for all 1 ≤ i ≤ n, εi = ±1.

Note that there are 2n n! s-permutations.

Theorem 1. Let X1, X2, . . . be real random variables with a joint distribution admitting adensity fn : Rn → R+ invariant under s-permutations.Then, for 0 ≤ m ≤ N ,

P [Nn = m] = (−1)n(−1

2

m

)(−1

2

n−m

).,

and P [Nn = m] = 0 for other values of m.

Here(zk

)is the generalized binomial coefficient, i.e. equal to z(z − 1) . . . (z − k + 1) / k!.

Before we prove Theorem 1, let us try and understand why an invariance under s-permutationsis asked for, and how the proof will rely on it. First, trivially:

P [Nn = m] = E [gn,m(X1, . . . , Xn)] =

∫Rn

gn,m(x1, . . . , xn) fn(x1, . . . , xn) dx1 . . . dxn

where gn,m : Rn → {0, 1} is the indicator function of the set {(x1, . . . , xn) ∈ Rn | (x1, . . . , xn) hasm positive partial sums}.Now, let us notice the following fact: given a point x = (x1, . . . , xn) ∈ Rn, the correspondingpartial sums — i.e. s0 = 0, s1 = x1, s2 = x1 + x2, etc. — may be written as the standard dotproducts between x and the vectors ηk, 1 ≤ k ≤ n, defined as n-tuples with 1s in the firstk components and only 0s afterwards. Therefore, the k-th partial sum of x will be positive ifand only if, in the Euclidean space Rn, x lies on the same side as ηk of the hyperplane Hηk

orthogonal to ηk. This is a simple geometric criterion for counting positive partial sums.

3

From the geometric criterion, we see that the law of Nn will be given by the integral of fnon regions delineated by the finite family of hyperplanes Hηk , because on such domains gn,mis identically 0 except for one value of m1. Hence, any choice of fn that leaves the probabilitycontents of these regions invariant will lead to the same law for Nn. Unfortunately, when n > 2,it is not the case that each of these regions may be built from a fundamental partition of thespace Rn into cells or chambers obtained from the full family of “coordinate” and “diagonal”hyperplanes: Hη, ηi = ±1 and 1 ≤

∑ni=1 η

2i ≤ 2. These hyperplanes generate the Weyl group

of the root system of type Bn, and the disjoint cells separated by the hyperplanes are calledWeyl chambers. There are 2n n! such chambers. When fn is invariant under s-permutations,the integral of fn on any two of the Weyl chambers is the same – thus, each Weyl chambercarries the same probability, viz. 1 / [2nn!].

Though gn,m is not constant on each Weyl chambers, it can be proved (and we did it inclass) that if we let

Gn,m(a1, . . . , an) =∑

εi=±1, σ∈Sn

gn,m(εσ(1)aσ(1), . . . , εσ(n)aσ(n)),

then Gn,m is constant in Rn \⋃

η∈{−1,0,1}n Hη. The constant value of Gn−m may be obtained

by looking at whichever point (a1, . . . , an) we like, and counting how many of its 2nn! images(including itself) under s-permutations have exactly m positive partial sums.

Lemma 1. Let a = (a1, . . . , an) ∈ Rn be such that ∀η ∈ {−1, 0, 1}n , η1a1 + · · · + ηnan 6= 0,except for η = 0. Then the number of s-permutations of (a1, . . . , an) for which exactly m partialsums are positive depends only on n and m, and is equal to

pn,m =n!

2n

(2m

m

)(2(n−m)

n−m

)= 2n n! (−1)n

(−1

2

m

)(−1

2

n−m

).

Proof. Let us pick a particular choice of a, one that satisfies

a1 > 0, a2 > a1, a3 > a1 + a2, . . . , an > a1 + · · ·+ an−1.

Thus the sign of a partial sum of an s-permutation of a is easily determined by just looking atthe sign of the term with highest index. Now, observe that one may obtain an s-permutation(x1, . . . , xn) of (a1, . . . , an) with m positive partial sums as follows. Pick m numbers aj1 , . . . , ajmthat will be the last terms (with plus or minus sign) in m positive partial sums. The remainingnumbers ajm+1 , . . . , ajn will be last terms in negative partial sums. There are

(nm

)possible

choices at this stage.If an is amongst aj1 , . . . , ajm then it must appear with positive sign in (x1, . . . , xn) since an >a1 + · · ·+ an−1. Then x1 + · · ·+xn > 0, and therefore |xn| is also one of the aj1 , . . . , ajm — andso possibly equal to an. If it is the case, then xn cannot be negative; but if |xn| 6= an, xn mightbe positive or negative. That is, 2m− 1 possibilities here.If an is not amongst aj1 , . . . , ajm , then |xn| is one of ajm+1 , . . . , ajn and there are similarly2(n−m)− 1 possibilities for xn.And so on for xn−1, xn−2, . . . , x1.Eventually, to fix a given s-permutation (x1, . . . , xn), one has chosen m times from aj1 , . . . , ajmand n−m times from ajm+1 , . . . , ajn . This leads to

pn,m =

(n

m

)(2m− 1) (2m− 3) · · · × 3× 1× (2(n−m)− 1) (2(n−m)− 3) · · · × 3× 1

= 2n n! (−1)n(−1

2

m

)(−1

2

n−m

).

1Note that the integral of fn on the union of the hyperplanes themselves will be 0 since we are integratingwith respect to the Lebesgue measure in Rn.

4

Using the Chu-Vandermonde identity, one checks easily that∑n

m=0 pn,m = 2n n!.

From Lemma 1 and the discussion immediately above it, it follows that

P [Nn = m] = (−1)n(−1

2

m

)(−1

2

n−m

),

as stated in Theorem 1.

Exercise: Show that, under the assumptions of Theorem 1,

limn→∞

P

(Nn

n≤ α

)=

2

πarcsin

√α,

for any α ∈ [0, 1].This generalizes a result established in 1947 by P. Erdos and M. Kac for random walks withindependent, identically distributed jumps having expectation 0 and variance 1 (Bull. Am.Math. Soc. 53, 1011-1020). As one might guess, the Arcsine law is also that of the sojourntime of a Brownian motion on the positive side of its origin (Levy, 1939).

5

2 Number of positive points & index of the maximum

As in the previous section, let X1, X2, . . . be real random variables with a joint distributionadmitting a density fn : Rn → R+ invariant under s-permutations. Let Nn be the number ofpositive points on the path of the random walk.From Theorem 1, observe that

P [Nn = m] = P [Nm = m]P [Nn−m = 0].

In words: the probability to have m positive points on the path is equal to the probabilityobtained when combining two independent subwalks, one with m steps and only positive pointsand one with n−m steps and only negative points! This is all the more surprising as this shouldnot be mistaken with the probability that the first m points of the walk be positive and theremaining n − m ones be negative — the number of positive points after the k-th step willindeed in general depend on the position Sk. The observation would in fact make sense moreeasily if instead of Nn we were considering a random variable for which there is some form ofindependence between the subwalk up to an arbirary step k and the subwalk after k. Thinkfor instance of the index of the maximal position: if you split the path at some step k, theindex of the maximal position in the subwalk up to step k is independent of the index of themaximal position in the subwalk after step k. The observation from Theorem 1 thus hints ata surprising and beautiful result, also established by Andersen. We state and prove it herefor independent, identically distributed jumps, but it is also valid for non-separable joint jumpdistribution invariant under s-permutations. It is not true for non separable symmetric jointjump distributions (i.e. invariant solely under permutations but not s-permutations).

Theorem 2. Principle of Equivalence Let X1, X2, . . . , Xn be independent, identically dis-tributed real random variables. Let Nn be the number of positive points on the correspondingrandom walk, and Kn be the index of the step when the maximal position is reached (thusSi < SKn for all i < Kn, and Si ≤ SKn for all i > Kn). Then, for any integer m,

P [Nn = m] = P [Kn = m].

Proof. In the same spirit as the proof of Theorem 1, we give a purely combinatorial proof. (Thisproof appears in W. Feller’s book, and was discovered by A.W. Joseph – who seems to havenot published it himself, although he did publish another proof of the same result, for which hecredits M.T.L. Bizley.) Given (a1, . . . , an) real numbers, let Ar be the number of permutationsof these numbers for which exactly r partial sums are positive (with 0 ≤ r ≤ n), and let Br bethe number of permutations for which the maximal value of the partial sums is (first) realizedby the rth partial sum.We prove by induction on n that Ar = Br for all 0 ≤ r ≤ n. Clearly, when n = 1, A0 = B0 andA1 = B1. Suppose the property holds for n − 1 ≥ 1. Let A

(k)r and B

(k)r be defined as Ar and

Br but solely on the (n− 1)-tuple obtained when removing ak from the list. By the induction

hypothesis, A(k)r = B

(k)r for all 0 ≤ r ≤ n− 1. Also, for r = n, A

(k)r = B

(k)r = 0. Now:

(i) if a1 + · · · + an ≤ 0, then An = Bn = 0, and for any permutation the number of positivepartial sums and the (first) index of the maximum will only depend on the first n− 1 elements.Since the n! permutations of (a1, . . . , an) may be partitioned according to which element is putin the last place, we therefore find that, for any 0 ≤ r ≤ n− 1,

Ar =n∑k=1

A(k)r and Br =

n∑k=1

B(k)r ,

which leads to Ar = Br by the induction hypothesis.(ii) if a1 + · · · + an > 0, then for any 1 ≤ r ≤ n one has that Ar =

∑nk=1 A

(k)r−1. For Br, let us

6

partition the set of all permutations of (a1, . . . , an) according to the element that is put in thefirst place. Then, the subwalk constructed with the n− 1 next elements reaches its maximumat its r-th step if and only if the full random walk (with the first element) reaches its maximumat step r+ 1. Indeed, when the first element is positive, this will just shift the subwalk upward,preserving the position of its maximum; and if the first element is non-positive, then it cannotbe “too” negative: if it were less than the opposite of the maximum of the subwalk, then thefinal position could not be positive, in contradiction with a1 + · · ·+ an > 0. Therefore, one alsohas that Br =

∑nk=1 B

(k)r−1. Hence, by the induction hypothesis, again Ar = Br for all r.

We leave the rest of the proof as an exercise for the reader.

Hence, to prove directly the observation made from Theorem 1, it now suffices to establishit for Kn.

Theorem 3. Let X1, X2, . . . , Xn be independent, identically distributed real random variables.Let Kn be the index of the step when the maximal position is reached (thus Si < SKn for alli < Kn, and Si ≤ SKn for all i > Kn). Then, for any integer m,

P [Kn = m] = P [Km = m]P [Kn−m = 0].

Proof. Note that by definition of Kn,

P [Kn = m] = P

[m−1⋂i=0

{Si < Sm} ∩n⋂

i=m+1

{Si ≤ Sm}

].

Hence

P [Kn = m] = P

[{Km = m} ∩

n⋂i=m+1

{Xm+1 + · · ·+Xm+i ≤ 0}

].

= P [{Km = m}]× P

[n⋂

i=m+1

{Xm+1 + · · ·+Xm+i ≤ 0}

]P [Kn = m] = P [Km = m]P [Kn−m = 0],

where we have used the independence of X1, . . . , Xn.

7

3 Edges on the convex hull of a planar random walk

We now move on to planar random walks, with a result that also relies on making use ofpermutations thanks to multiple symmetries.

Let us consider vectors in the plane: z1, . . . , zn ∈ R2. Let z = z1 + . . . zn, and for 1 ≤ k ≤ n, wealso write as before the partial sums sk = z1 + · · ·+ zk. We will not need here s-permutationsbut only standard permutations, and for σ ∈ Sn we write

sk(σ) = zσ(1) + · · ·+ zσ(k).

For A ⊂ 1, . . . , n, we let zA =∑

i∈A zi. We also introduce a notion that plays the same rolehere as the condition in Theorem 1 in the first section.

Definition 2. The family of vectors z1, . . . , zn is said to be skew if:

∀A,B ⊂ {1, . . . , n}, zA = zB ⇒ A = B.

Under this condition, for a given choice of A ⊂ 1, . . . , n, the number of permutations forwhich zA appears as an edge on the boundary of the convex hull of the path {0, sσ(1), . . . , sσ(n)}depends only on n and the cardinal of A, as we show now.

Lemma 2. Let z1, . . . , zn be skew. Then

1. There exists a unique cyclic permutation ρ ∈ Sn such that the partial sums sρ(1), . . . , sρ(n1)

all lie on the same side of the line with direction z.

2. Let A ⊂ 1, . . . , n with m elements. Then zA appears as an edge on the boundary of theconvex hull for exactly

2 (m− 1)! (n−m)!

of the n! paths {0, sσ(1), . . . , sσ(n)} obtained as σ ranges over Sn.

Proof. 1. z1, . . . , zn skew ⇒ there is at most one point among {0, s1, . . . , sn−1} which liesat a maximum distance from the line with direction z. Say this point is sk. Then take:

ρ(1) = k + 1, ρ(2) = k + 2, . . . , ρ(n− k) = n, ρ(n− k + 1) = 1, . . . , ρ(n) = k.

This is a cyclic permutation of z1, . . . , zn, with all partial sums on the same side of the linewith direction z. The uniqueness of σ thus defined follows from that of k – and thereforefrom the skewness of z1, . . . , zn.Note: if one quotients Sn by the equivalence relation corresponding to cyclic permutations(i.e. two permutations σ1, σ2 are in the same equivalence class if and only if there existsa cyclic permutation ρ such that σ1 = ρ ◦ σ2), the result above means that there is onlyone permutation per class for which all points lie on the same side of the line given by z.Hence there are in total (n− 1)! permutations σ such that sσ(1), . . . , sσ(n1) lie on the sameside of the line (since there are n permutations in each class, and of course n! elementsin Sn).

2. Let zn+1 = −sn = −z and, given A ⊂ 1, . . . , n with m elements, let

A′ = {1, . . . , n, n+ 1} \ A.

Also, sn+1(σ) = s0(σ) = 0 for all σ ∈ Sn.Note that zA appears as an edge on the boundary of the convex hull of {sσ(0), sσ(1), . . . , sσ(n)}

8

if and only if there exists 0 ≤ k ≤ n−m such that zA = sk+m(σ)− sk(σ) and all partialsums of {zσ(k+1), . . . , zσ(k+m)} lie on the same side of the line given by zA and all partialsums of {zσ(k+m+1), . . . , zσ(n), zn+1, zσ(1), . . . , zσ(k)} lie also on that same side of the line.From the first point, we have that there will be 2 × (m − 1)! × (m − n)! permutationsfor which such a configuration happens. (The factor 2 comes from the fact that there aretwo sides to a line.)

To apply the preceding lemma to a planar random walk, note that if Z1, . . . , Zn are inde-pendent, identically distributed random vectors all with a density with respect to the Lebesguemeasure, then almost surely they form a skew family.

Definition 3. For Z1, . . . , Zn independent, identically distributed random vectors, define

Cn = conv (0, S1, . . . , Sn) ,

the convex hull of the corresponding random walk, and ∂Cn its boundary. Note that almostsurely ∂Cn is a finite union of edges (straight line segments).Let Kn = Card ({i | Zi ∈ ∂Cn}), where by Zi ∈ ∂Cn we mean that the vector Zi is an edge onthe boundary of the convex hull; Vn = Card ({ edges of ∂Cn}) = Card ({ vertices on ∂Cn}); andLn = Length of the perimeter of Cn = Length of ∂Cn.

Letting Wn be any of Kn, Vn or Ln and writing Wn(σ) for Wn computed on the random walkconstructed from Zσ(1), . . . , Zσ(n), one finds that

E

[∑σ∈Sn

Wn(σ)

]= n!E[Wn] ,

as Z1, . . . , Zn are independent, identically distributed random vectors.2

Now we are ready to prove the following theorem.

Theorem 4.

E[Kn] = 2

E[Vn] = 2n∑k=1

1

k

E[Ln] = 2n∑k=1

E [ ‖Sk‖ ]

k

Proof. Exercise

The previous theorem is in fact a special case of the next one, with a similar proof.

Theorem 5. Writing l1, . . . , lVn for the lengths of the edges on ∂Cn (numbered eg counter-clockwise from the one intersecting a preferred half-line from the origin), we define for any realvalued function f the random variable Fn =

∑Vnk=1 f(lk). Then, one has that

E[Fn] = 2n∑k=1

E [f(‖Sk‖)]k

.

2In fact, we see here that it is sufficient to have Z1, . . . , Zn exchangeable random vectors, each with at leastone of their components admitting a density (so that they form a.s. a skew family).

9

4 Some combinatorial lemmas in higher dimensions

One way to extend the results in the previous section to higher dimensional random walks isto establish combinatorial lemmas similar to the one on which these results rely.

Lemma 3. Let z1, . . . , zn be a family of p-dimensional vectors (n ≥ p). Let H be a hyperplanewhich contains z = z1 + · · ·+zn but no other sum of any subset of the family. Let also C denoteone of the half-spaces determined by H. Then, there exists exactly one cyclic permutation ρsuch that exactly m of the partial sums s1(ρ), . . . , sn−1(ρ) lie in C.

Proof. Let u be a unit vector normal to H. There exists exactly one index k such that thescalar product n.sk is minimal. Then simply set ρ(1) = k+ 1, . . . , ρ(n− k) = n, ρ(n− k+ 1) =1, . . . , ρ(n) = k.And the uniqueness of ρ follows from that of k.

Before the next lemma, we extend further the one-dimensional notion of skewness for afamily of vectors.

Definition 4. We say that z1, . . . , zn forms a skew family if and only for any non-empty disjointsubsets A1, A2, . . . , Ap of 1, . . . , n the vectors zA1 , . . . , zAn are linearly independent. (Recall thatzAj =

∑i∈Aj zi.)

Lemma 4. Let A1, A2, . . . , Ap−1 be non-empty disjoint subsets of 1, . . . , n. Then the vectorszA1 , . . . , zAp−1 appear on the boundary of the convex hull of exactly

2 (nA1 − 1)! (nA2 − 1)! . . . (nAp−1 − 1)! (n− nA1 − · · · − nAp−1)!

paths out of the n! paths s0(σ), s1(σ), . . . , sn(σ) as σ ranges over all permutations.

Proof. Exercise.

Exactly in the same way as the formula for the number of edges was obtained in the previoussection, we derive from the lemma above the following higher-dimensional formula for faces onthe boundary of the convex hull of a random walk with iid steps.

Theorem 6. Let Z1, . . . , Zn be independent, identically distributed random vectors in Rd andlet Vn be the number of faces on the boundary of the convex hull of the corresponding randomwalk. Note that, a.s., a face is a d-1-dimensional simplex. Then

E[Vn] = 2∑

k1+···+kd−1≤nk1,...,kd−1≥1

1

k1 · k2 · · · kd−1

∼n→∞ 2 [log(n)]d−1

10

5 A different approach to the perimeter:

the Spitzer-Widom formula

Here we follow a different path to compute the expected perimeter length of the boundary ofthe convex hull of a random walk. The idea is based on a formula established by Cauchy:

Lemma 5. Let A be a set of points in the plane, and L be the length of the boundary of theconvex hull of A. Then

L =

∫ π

0

D(θ) dθ,

where D(θ) = maxz∈A(x cos θ + y sin θ)−minz∈A(x cos θ + y sin θ)

Proof. Exercise.

Cauchy’s formula essentially enables one to reduce a two dimensional problem to a onedimensional one: that of knowing the distribution of the diameter or the range, i.e. the quantitymaxz∈A(x cos θ+ y sin θ)−minz∈A(x cos θ+ y sin θ), for the set under consideration. One couldthink that results would then depend on the specifics of the jump law, but in fact anothercombinatorial lemma will lead again to a general formula with wide applicability, provided thesteps of the random walk are independent and identically distributed.

Lemma 6. Let a1, . . . , an be real numbers. For σ ∈ Σn a permutation and 1 ≤ k ≤ n, write asbefore sσ(k) = aσ(1) + · · ·+ aσ(k). Then∑

σ∈Σn

max{

0, sσ(1), . . . , sσ(n)

}=∑σ∈Σn

Nn(σ) aσ(1),

where Nn(σ) is the number of positive elements in {sσ(1), . . . , sσ(n)}.

Proof. Notice that for any 1 ≤ k ≤ n,

max{

0, sσ(1), . . . , sσ(k)

}−max

{0, sσ(1), . . . , sσ(k−1)

}=

1{sσ(k)>0}(aσ(1) + max

{0, sσ(2) − sσ(1), . . . , sσ(k) − sσ(1)

}−max

{0, sσ(1), . . . , sσ(k−1)

}).

Choosing i1, . . . , ik, one may decompose Σn into subsets of permutations that send {1, . . . , k}onto {i1, . . . , ik}. Summing over these and then over all choices of i1, . . . , ik leads to cancellationsof all terms but 1{sσ(k)>0} aσ(1). Finally summing over 1 ≤ k ≤ n finishes the proof.

From the previous lemma, one may deduce the following theorem:

Theorem 7.∑σ∈Σn

[max

{0, sσ(1), . . . , sσ(n)

}−min

{0, sσ(1), . . . , sσ(n)

}]= 2

∑σ∈Σn

n∑k=1

1

k|sσ(k)|,

A straightforward corollary of which is the Spitzer-Widom formula for the expected perime-ter length of the convex hull of a planar random walk with iid steps:

E[Ln] = 2n∑k=1

E [ ‖Sk‖ ]

k.

11

6 Digression: Brownian meanders and excursions

Another way of looking at Baxter’s vertex formula, E[Vn] = 2∑n

k=11k, is to think of the

associated combinatorial lemma as implying the following: consider the line segment joiningtwo points of the random walk separated by k steps, then the probability that it be an edge onthe convex hull of the random walk is 2/[k

(nk

)]. If one wants to find a (form of) continuous limit

of Baxter’s formula, say for the convex hull of a Brownian motion, one needs a few preliminarynotions: crossing a line for a continuous time process such as Brownian motion is somethingquite different from crossing a line for a discrete time random walk. The tools needed are thoseassociated with so-called Brownian meanders and excursions.

In his seminal work on Brownian excursions K.L. Chung surveys variants of Brownian motionconditioned to stay positive. For a standard Brownian motion Bt, the excursion straddling tand the meandering process ending at t may be defined from the last zero before t and the firstzero after t, namely:

γ(t) = sup {s ≤ t, Bs = 0}and

β(t) = inf {s ≥ t, Bs = 0} .Then the absolute value of the process between γ(t) and β(t), conditional on these, is theexcursion straddling t. Thus an excursion starts from 0 and returns to 0 but without hitting 0between its initial and final points (see Fig. 1). If one performs conditioning simply on γ(t)and focuses on the pre-t part, one obtains a meandering process: a process starting at 0 andnever hitting 0 again on the time interval [γ(t), t].

Figure 1: Brownian excursion: Brownian motion conditioned to stay positive and to return to 0.

Excursions and meanders may similarly be defined for general Levy processes. In the caseof Brownian motion, the propagator3 for the meander is known to be [18]

g(x; t) =x

te−

x2

2t . (1)

Recalling that the first passage-time density for a Brownian motion, that is the probabilitydensity associated with the first zero of a Brownian motion starting at x > 0 is

ρ(x, t) =x√2πt3

e−x2

2t , (2)

and that the standard Brownian propagator from 0 to 0 in time t is

f(0, 0; t) =e−(0−0)2/2t

√2πt

=1√2πt

, (3)

3 I use here more of a physicist’s vocabulary. The propagator is the same as the transition density of the stochastic process.

For instance the propagator of standard Brownian motion is f(x, y; t) = e−(y−x)2/2t√

2πt.

12

one notices that:

g(x; t) =ρ(x, t)

f(0, 0; t). (4)

This is in fact not a mere coincidence. A simple path transformation accounts for this obser-vation and allows one to establish the same result for all symmetric α-stable processes, as weshall see below.

6.1 Path transformations

A well known example of a path transformation is the so-called method of images, whichaccording to Feller is due to Lord Kelvin. This transformation allows one to compute thepropagator from x > 0 to y > 0 for a Brownian motion constrained to stay positive:

q(x, y; t) =1√2πt

(e−

(y−x)2

2t − e−(y+x)2

2t

). (5)

Indeed, the method of images is equivalent to transforming paths that go from x > 0 to y > 0and hit 0 into paths that go from −x to y. The bijection thus established enables one toidentify the right factor to subtract from the standard Brownian density in order to obtain theconstrained propagator (Eq. 5).

Another example of path transformation is Vervaat’s construction of a Brownian excursionfrom a Brownian bridge.

Figure 2: Path transformation leading from a Brownian bridge with minimum −x/2 (top) to a Brownian meander with finalvalue x > 0 (bottom). The pre-minimum part is time-reversed and shifted upward by x/2 while the post-minimum part is shiftedupxard by x.

Let us describe now the path transformation we wish to use here. We start with a Brownianmotion B of duration t > 0. The probability density function (pdf) for such a motion to beactually a Brownian bridge (that is, B0 = Bt = 0) is given by the Brownian propagator from 0to 0 in time t:

f(0, 0; t) =e−(0−0)2/2t

√2πt

=1√2πt

. (6)

13

We constrain B to be a Brownian bridge and we let τm be the time when B attains its minimumon the interval [0, t] – note that this minimum is non-positive. Consider now the pre-τm andthe post-τm parts. For s ∈ [0, τm], define:

W+s = Bτm−s −Bτm , (7)

and for s ∈ [τm, t]:W+s = Bs − 2Bτm . (8)

That is, the pre-τm part is time-reversed and the post-τm part is shifted to be appended to thenew path, as shown in Fig. 2. Recall that B(0) = 0 and note that B(τm) ≤ 0.This path transformation produces a Brownian meander W+ of duration t, with final point x >0, where −x/2 is the minimum of the initial Brownian bridge. This transformation first ap-peared in the literature in a 1991 paper by Jean Bertoin.

6.2 Propagator for the Brownian meander

The path transformation defined in Eq. 7 and 8 means that the propagator for a Brownianmeander with terminal point x > 0 is the same as the pdf for the minimum of a Brownianbridge evaluated at −x/2. (Both meander and bridge having same duration t.) The latter is

well known [21] to be xte−

x2

2t and thus one obtains directly the propagator of the meander, asin Eq. 1.One may also, thus, arrive directly at Eq 4 and understand why this equality is not a merecoincidence. Indeed, since the path transformation is a bijection, it implies that the totalprobability density associated with Brownian meanders of duration t (whatever their endpoint)is the same as that associated with Brownian bridges of duration t (whatever the value oftheir minimum). In other terms, the partition function for Brownian meanders of duration tis f(0, 0; t) = 1√

2πt. And the numerator to be set against this partition function if one wishes

to compute the propagator of a meander ending at x > 0 is simply the pdf associated with astandard Brownian path starting from x and first hitting 0 at time t – namely the first-passagetime density ρ(x, t). Hence Eq. 4:

g(x, t) =ρ(x, t)

f(0, 0; t).

Incidentally, this also explains why∫ ∞0

ρ(x, t) dx = f(0, 0; t); (9)

an equality which, though easily checked in the Brownian case, could have been mistaken aspurely coincidental, whereas it reflects the bijection between meanders and bridges.

6.3 Generalization to other Levy processes

The path transformation introduced above relies only on properties of Brownian motion thatare actually valid also for symmetric stable processes, most importantly Markovian and time-reversal properties. The same result therefore holds – more precisely, one needs to be carefulwith the jumps that appear in stable processes, and it is better in this case to append thetime-reversed pre-minimum part after the post-minimum one [6, 13, 14, 16]. This leads to newequalities on the spatial integrals of first-passage-time densities for symmetric stable processes.

14

For instance in the case of a Cauchy process (i.e. a symmetric α-stable process with α = 1),one finds that: ∫ ∞

0

ρ(x, t) dx =1

πt. (10)

And also, more generally, for a symmetric α-stable process:∫ ∞0

ρ(x, t) dx = f(0, 0; 1)t−1α . (11)

Thus, thanks to a path transformation, one obtains identities relating probability densityfunctions that, but in a few cases, are not known explicitly.

15

7 Edges on planar Brownian convex hulls

Consider a planar Brownian motion B(t) = (b1(t), b2(t)), that is, a random variable in the planecorresponding to the position of a Brownian particle in some canonical reference frame. Thecoordinates b1 and b2 are thus two independent copies of standard, one-dimensional Brownianmotion. We call Γ(T ) the path described in the plane by the Brownian particle up to time T ,

Γ(T ) = {(b1(s), b2(s)) , 0 ≤ s ≤ T} . (12)

C(T ) is the convex hull of Γ(T ), i.e. the smallest convex set in the plane containing all pointsof the path at time T , and we write ∂C(T ) for the boundary of C(T ). Let us recall that ∂C(T )is (almost surely) a countable union of straight line segments [22, 23, 19].

Strictly speaking the number of edges is countably infinite, so we use a regularization strat-egy to analyze its behaviour and derive an explicit formula. The (simple) idea is to computean “edge probability”. Namely, picking two points on the path, B(τ), B(τ + s) ∈ Γ(T ), what isthe probability that the line segment [B(τ), B(τ + s)] is an edge on the boundary of the convexhull? To be more precise, let us write

Aτ,τ+s = {[B(τ), B(τ + s)] ⊂ ∂C(T )} . (13)

The edge probability is thus simply the probability of the event Aτ,τ+s. This probability maybe written as the expectation of the indicator function of the event:

P (Aτ,τ+s) = E[IAτ,τ+s

]. (14)

So that, somewhat naively, the average number N (T ) of edges on ∂C(T ) may be written usingthe following, formal summation:

“ N (T ) = E

∫

∆t≤ s≤T0≤ τ ≤T−s

IAτ,τ+s

=

∫∆t≤ s≤T

0≤ τ ≤T−s

P (Aτ,s) ” (15)

with ∆t a regularization parameter (akin to a time separation below which we do not considerline segments).To make this rigorous, we need to compute a bivariate density ρ(τ, s) associated with P (Aτ,τ+s),

Figure 3: A planar Brownian path. The dashed lines are coordinate axes in a canonical reference frame. We have representedthe boundary of the convex hull: that is, the boundary of the smallest convex set enclosing all points of the path.

16

and we shall see later on that this is indeed a bona fide joint density4. N (T ) will then take thefollowing form:

N (T ) =

∫∆t≤ s≤T

0≤ τ ≤T−s

ρ (τ, s) dτ ds. (16)

Figure 4: Two possible configurations of the Brownian path Γ(T ), with respect to the line joining B(τ) and B(τ + s).

Figure 5: The LHS configuration of Fig. 4 shown in an adapted reference frame (a), and the (new) y-coordinate plotted withrespect to time (b).

The key step is thus to compute the edge probability P (Aτ,τ+s). This may be done byshifting to a reference frame adapted to the straight line segment under consideration, as weexplain below.

Given times s ∈ [∆t, T ] and τ ∈ [0, T − s], Γ(T ) may be found either:

• to lie on one side only of the line through B(τ) and B(τ + s), as in Fig.4 (a)

• or, to cross this line, as in Fig.4 (b).

The adapted reference frame we consider is one where the line through B(τ) and B(τ +s) istaken as the x-axis, as shown in Fig. 5. The “edge” event Aτ,τ+s thus translates into constraintsfor the (new) y coordinate of the motion, viz.

C1. y(t) starts at y0 > 0 (say) and hits 0 for the first time at t = τ ;

C2. y(t) hits 0 again at t = τ + s but remains strictly positive between t = τ and t = τ + s;

C3. y(t) does not hit 0 after time t = τ + s, it remains strictly positive up to time t = T .

4 We shall see that ρ(τ, s) may be viewed as the joint density of the time of the global minimum of a Brownian motion withduration T − s, and the sojourn time in R+ of a Brownian bridge with duration s.

17

Thanks to the Markovian nature of Brownian motion, we can separate the path into threeparts and treat them independently. The constraints C1, C2, C3 mean we are then dealingwith subpaths that are Brownian meanders and Brownian excursion5. We refer the readerto [33] for the full details of the calculation, in particular that of the normalization constants.Let us simply mention here the expression thus obtained for the edge probability:

P (Aτ,τ+s) =2 dτ ds

πs√τ [(T − s)− τ ]

, (17)

that is,

ρ (τ, τ + s) =2

πs√τ [(T − s)− τ ]

. (18)

The most interesting thing in this expression is the probabilistic interpretation that may begiven to its factors. Indeed, 1

π√τ [(T−s)−τ ]

× 1s

is nothing but the product of the uniform density

on an interval of length s (that is, 1/s), and an arcsine density on the interval [0, T − s] (thatis, 1/π

√τ [(T − s)− τ ]). This may be understood by “cutting and stitching” the y-coordinate

path in the adapted reference frame shown in Fig. 5 (b). Treating the middle part (between τand τ +s) separately, one has a Brownian excursion or, seen otherwise, a Brownian bridge witha sojourn time on the positive side equal to its full duration, s. The corresponding sojourntime density is well-known to be uniform [29] and this leads to the 1

s-factor. As for the two

end parts of the path, they can be concatenated to produce a Brownian path of duration T − sthat hits its minimum at τ . The corresponding density is the well-known arcsine one, derivedby Paul Levy [30] – this gives the 1

π√τ [(T−s)−τ ]

-factor. The overall factor of 2 comes from the

fact that there are two sides to a straight line.Although, in the Brownian case, the edge probability can be computed by “brute force”

(one is, after all, simply talking about Gaussian integrals), the path transformation approachis much more appealing – and not only to those of us who dislike long calculations! Indeeda path transformation corresponds to a bijection between sets of paths, and chances are thatit generalizes from Brownian motion to more general Markov processes. Path transformationsare commonplace in the study of Brownian motion and Levy processes [40, 10, 5, 6, 8, 14, 15,1, 17, 12, 7, 31].

For now, let us remark that the “cutting and stitching” described above, together with thearcsine law for the time of the minimum of a Brownian motion and the uniform law for thesojourn time of a Brownian bridge, leads immediately to Eqs. (17) and (18). From which,upon integration of ρ (τ, τ + s) over τ , one finds∫ T−s

0

ρ (τ, τ + s) dτ =2

s, (19)

because ρ (τ, τ + s) depends on τ only through the arcsine density on [0, T − s], so that theintegral over τ ∈ [0, T − s] yields 1. Then, to compute the average number of edges N (T ) fromEqs. (16) and (19), all that is left is to integrate over s ∈ [∆t, T ], leading to

N (T ) = 2 ln

(T

∆t

)(20)

This result generalizes the discrete-time (random walk) result [4], through the idea of computingedge probabilities. But before presenting higher-dimensional results, let us mention some othertwo-dimensional results derived using edge probabilities.

5 Let us recall that: (i) a Brownian bridge is a Brownian motion that starts at 0 and is constrained to come back to 0, (ii) aBrownian excursion is a Brownian bridge further constrained to always keep the same sign (equivalently, it is a Brownian motionconstrained to always stay positive and come back to 0), (iii) a Brownian meander is a Brownian motion that starts at 0 and isconstrained to stay positive (but it does not have to come back to 0). See Section ??.

18

8 Convex hull of multiple planar Brownian paths

Figure 6: Two types of edges on the convex hull of n = 9 independent planar Brownian paths (with same origin and sameduration): edges joining points on the same path (a), edges joining points on two different paths (b).

Indeed, long but straightforward computation of Gaussian integrals allows one to obtainthe average number, Nn(T ), of edges on the boundary of the convex hull of n ≥ 1 independentBrownian paths. One has to consider the two types of edges that can exist: edges joining twopoints on the same Brownian path, and edges joining two points on two different paths (Fig. 6).In the relevant adapted reference frame, this induces two types of configurations for the y-coordinates, as shown in Fig 7. Computing the associated densities leads to a closed expression

Figure 7: One-dimensional (y-coordinate) configurations corresponding to the two types of edges on the convex hull of n = 9independent planar Brownian paths.

for Nn(T ) (this was done in the second part of [33] and we skip details here):

Nn(T ) = αn(T,∆t) + βn(T,∆t) (21)

with

αn(T,∆t) =

∫ 1

∆tT

∫ 1−v

0

∫ ∞0

h(a)n (u, v, z) du dz dv,

(22)

βn(T,∆t) =

∫ 1

∆tT

∫ min(1,v)

max(0,v−1)

∫ ∞0

h(b)n (u, v, z) du dz dv,

(23)

19

where

h(a)n (u, v, z) = 4n

u exp(−u2

z

)[erf(u)]n−1

πvz [z(1− v − z)]12

(24)

h(b)n (u, v, z) = 4n(n− 1)

v12u2 exp

(− vu2

z(v−z)

)[erf(u)]n−2

πz(v − z) [πz(1− z)(v − z)(1− v + z)]12

(25)

(We write erf(u) = 2√π

∫ u0e−x

2dx for the error function.)

These expressions, though maybe not very much pleasing to the eye, are an example of ex-act analytical results that may be obtained using edge probabilities in a context where theunderlying probability densities are tractable (typically, Gaussian densities).

20

9 Edges on planar Levy convex hulls

Now, let us come back to single path contexts. As mentioned above, the path transformationthat leads directly to the edge probability may be used to compute the average number of edgeson more general Levy convex hulls, eg convex hulls of symmetric α-stable Levy processes otherthan simple Brownian motion. Recall Eq.17:

P (Aτ,τ+s) = E[IAτ,τ+s

]=

2 dτ ds

πs√τ [(T − s)− τ ]

.

Recall also that the 1

π√τ [(T−s)−τ ]

- factor corresponds to the two end parts of the path: once

stitched together, these amount to a process of duration T −s that attains its minimum at timeτ . Last but not least, recall that the edge probability is integrated over τ ∈ [0, T − s]. So that,in fact, one does not even need to know the specific form of the underlying marginal density(the arcsine density in the Brownian case): it is going to be integrated out and yield 1 in anycase. Thus the only important factor in the edge probability is 2

s. Where did this factor come

from? From the middle part of the process, the one that corresponds to an excursion awayfrom the edge. Such an excursion may also be viewed as a bridge with a sojourn time in thepositive (or negative) side equal to its full duration. As it happens, the sojourn time densityis uniform not only for Brownian bridges, but also for bridges of a large class of cyclicallyexchangeable processes [24, 28], including for instance most symmetric α-stable processes6.These so-called uniform laws are the continuous-time equivalent of combinatorial identities forrandom walks [34, 26, 35, 36, 37, 38, 4]. They emerge from the profound symmetry of cyclicallyexchangeable processes and are intimately linked to Ballot theorems [27] (we shall encounterBallot theorems [43, 9, 3, 2, 39] also in the second part of this thesis, see Section ??.)

Figure 8: Computer simulation results (points) compared with the exact formula from Eq. (26) (solid lines) giving the averagenumber of s-edges with s ∈ [∆τ1,∆τ2], for a range of values of ∆τ2 and ∆τ1 (∆τ1 = 10−1 on the bottom curve, 10−2 on thesecond curve, 10−3 on the third one and 10−4 on the top curve). Symbols correspond to computer simulations with 104 paths for:Brownian motion with diffusion constant set to 1/2 (filled squares) or 50 (filled circles), Cauchy-Lorentz process (filled triangles),symmetric α-stable processes with α = 0.75 (squares) and α = 1.5 (circles), compound Poisson processes with drift and with normal(pluses) or Cauchy (crosses) jump variables.

Therefore, one obtains the same logarithmic form as in Eq. 20 for the general symmetricLevy case. This form may be further generalized as to express the average number of edgesjoining points with a time-separation in [∆t1,∆t2]

N[∆t1,∆t2](T ) = 2

∫ ∆τ2

∆τ1

1

sds = 2 ln

(∆τ2

∆τ1

)(26)

6 The Fourier transform of the process needs to be integrable.

21

This formula shows the remarkable “universal” behaviour of the average number of edgeson the convex hull of symmetric processes, as well as a time-scale invariance: there are asmany edges joining points separated by δt ∈ [∆t1,∆t2] as edges joining points separated byδt ∈ [λ∆t1, λ∆t2] (for ∆t

∆t1≤ λ ≤ T

∆t2). Numerical simulations illustrate the validity of this

general formula for various symmetric Levy processes, as can be seen in Fig. 8.The universality observed for the average number of edges in two dimensions extends to the

average number of facets in d ≥ 3 dimensions (Fig.9), as we shall see in the next section.

22

10 Convex hulls of higher-dimensional Levy processes

Figure 9: A sample realization of 3-dimensional Brownian motion and the facets on the boundary of its convex hull.

Moving to higher dimensions, edges become faces or facets and the same idea of computing“facet probabilities” leads to an exact, integral formula for the average number of facets on theconvex hull of a vast class of higher-dimensional Levy processes. In particular, in dimensiond = 3, the integral formula for the average number of facets admits a closed form,

N (3)(T ) = 2 [ln (T/∆t)]2 + 4{

ln (T/∆t) ln (1−∆t/T )− Li2 (1−∆t/T ) + π2/12},

where Li2 is the dilogarithm function, T is the total duration of the process and ∆t the usual cut-off parameter representing an arbitrary minimal time separation between two extreme pointson the process’s path. The asymptotics in the general d-dimensional case is

N (d)(T ) ∼ 2 [ln (T/∆t)]d−1 , (27)

which nicely generalizes the two-dimensional case.

In dimension d, a facet will be a (d−1)-dimensional object lying in a hyperplane and definedby d points on the process’s path. One thus splits the path into d + 1 independent parts, andthen computes the probability density (pdf) associated with each part before multiplying themtogether and integrating. When d = 3, the four parts and their associated pdfs are:

• from time 0 up to time τ , the path lies on one side only (say side +) of a (hyper)plane

H, and it hits H at time τ ; write p(0)τ for the corresponding pdf;

• from time τ up to time τ + k1, the path is an excursion away from H (in side +) and ispinned on H at times τ and τ + k1; write f(τ, τ + k1) for the pdf;

• from time τ +k1 up to time τ +k1 +k2, the path is an excursion away from H (in side +)and is pinned on H at times τ + k1 and τ + k1 + k2; write f(τ + k1, τ + k1 + k2) for thepdf;

• from time τ + k1 + k2 up to time T , the path lies on side + of H and it is pinned on Hat time τ + k1 + k2; write p

(τ+k1+k2)T for the pdf.

Formally, one obtains the following integral

N (3)(T ) = 2

∫ T−∆t

∆t

∫ T−k1

∆t

∫ T−(k1+k2)

0

p(0)τ f(τ, τ +k1)f(τ +k1, τ +k1 +k2)p

(τ+k1+k2)T dτ dk2 dk1.

(28)

23

Since Levy processes have stationary and independent increments, we have, for all τ , k1

and k2: f(τ, τ + k1) = f(0, k1) ≡ f(k1) and f(τ + k1, τ + k1 + k2) = f(0, k2) ≡ f(k2). Also,

the product p(0)τ p

(τ+k1+k2)T can be interpreted as giving the pdf of the time τ at which a similar

process of duration T − (k1 + k2) attains its minimum. Thus,∫ T−(k1+k2)

τ=0

p(0)τ p

(τ+k1+k2)T dτ = 1, (29)

and the triple integral in Eq. (28) reduces to

N (3)(T ) = 2

∫ T−∆t

∆t

∫ T−k1

∆t

f(k1)f(k2) dk2 dk1. (30)

Lastly, f(k) may be viewed as the pdf that the sojourn time of a process on a given sideof a (hyper)plane to which it is pinned (at both endpoints) is equal to the full duration k.This means, f(k) is the pdf of the sojourn time of a Levy bridge. Direct higher dimensionalresults [11, 20] or the same reasoning as before (i.e. using an adapted frame and reducing theproblem to a one-dimensional one) shows that the sojourn-time distribution is uniform for avast class of Levy processes [24, 28, 27]. Therefore f(k) = 1/k, and we obtain

N (3)(T ) = 2

∫k1+k2≤Tk1,k2≥∆t

dk1 dk2

k1k2

. (31)

In dimension higher than 3, Eq. (28) becomes a d-fold integral, accounting for facets definedby d points and characterized by d− 1 excursions:

N (d)(T ) = 2

∫ T−(d−2)∆t

∆t

∫ T−(d−3)∆t−k1

∆t

· · ·∫ T−(k1+···+kd−2)

∆t

dkd−1 · · · dk2 dk1

k1k2 · · · kd−1

,

that is,

N (d)(T ) = 2

∫k1+···+kd−1≤Tk1,...,kd−1≥∆t

dk1 · · · dkd−2 dkd−1

k1k2 · · · kd−1

. (32)

In the three-dimensional case, it is possible to obtain a closed form for this integral formula, asdetailed in the next paragraph.

Closed formula and asymptotics

Starting from Eq. (31), we change variables to y1 = k1, y2 = k1 + k2. We find

N (3)(T ) = 2

∫ T−∆t

∆t

dy1

y1

∫ T

y1+∆t

dy2

y2 − y1

, (33)

in which the second integral is easily computed. This leads to:

N (3)(T ) = 2

∫ T−∆t

∆t

dy1

y1

[ln

(T

∆t

)+ ln

(1− y1

T

)]. (34)

Setting z = y1/T allows one to compute the remaining integral (see also [32, 44]) and yieldsthe announced result,

N (3)(T ) = 2 [ln (T/∆t)]2 + 4{

ln (T/∆t) ln (1−∆t/T )− Li2 (1−∆t/T ) + π2/12}, (35)

24

1

10

20

30

40

1 20 50 80 100

<F

T>

T / ∆t

100

300

103

104

105

Figure 10: Numerical simulations for the average number of triangular facets on the convex hulls of some 3-dimensional Levyprocesses (with 104 and 103 sample paths’ realizations): standard Brownian motion (pluses), and a stable process with stabilityindex 1.5 (triangles). The solid line corresponds to the exact formula Eq. (35). The inset shows the asymptotics for large values ofT/∆t, from Eq (36), with numerical simulations of 3-dimensional Brownian motion.

where Li2 is the dilogarithm function. As Li2 (1−) is finite, the first term on the right hand sideof Eq.(35) is also clearly the leading term when T/∆t becomes large. Thus,

N (3)(T ) ∼ 2 [ln (T/∆t)]2 . (36)

This asymptotical behaviour may readily be directly extracted from Eq. (34), focusing on theln (T/∆t) term in the integrand. This transposes straightforwardly, by simple iteration, to the(d− 1)-dimensional integral in Eq (32), leading to:

N (d)(T ) ∼ 2 [ln (T/∆t)]d−1 . (37)

This result generalizes in a very simple manner the two-dimensional one as well as the discrete-time one. The exact formula (Eq. (35)) and the asymptotical behaviour (Eq. (36)) are illustratedin numerical simulations, see Fig. 10.

Once again, the exact formula established here for the average number of (d−1)-dimensionalfacets on the convex hull of a general d-dimensional Levy process shows how universal certainaspects of the shape of spatial random processes are. Indeed this formula is valid not onlyfor d-dimensional Brownian motion but for a vast class of Levy processes, including all stableprocesses. This is reminiscent of the universality observed in the Sparre Andersen theorem [35,36], to which the formula established here is linked via the arcsine and the uniform laws.

25

11 Multidimensional Random Walks

We return to discrete-time settings and we elaborate here on a paper by Vysotsky and Za-porozhets [41]. Recall Sparre Andersen’s famous result for the persistence probability of a1-dimensional random walk with continuous, symmetric joint distribution of increments:

P (S1 > 0, . . . , Sn > 0) =(2n− 1)!!

(2n)!!.

In dimension d ≥ 2, instead of positivity, one may look at the probability that the origin be“absorbed” in the interior of the convex hull. A result close to that, only for independent pointsrather than positions of a random walk, was established by Wendel [42]:

P (0 /∈ Conv(X1, . . . , Xn)) =1

2n−1

d−1∑k=0

(n− 1

k

),

provided X1, . . . , Xn are i.i.d. with a centrally symmetric distribution and such that P (X1 ∈H) = 0 for any linear hyperplane H.

In this section we focus on random walks with exchangeable increments and we shallmainly make regular use of three more assumptions:

(H) P (X1 ∈ H) = 0 for any affine hyperplane H;

(G) P (S1, . . . , Sn in general position) = 1, where “in general position” means that any dvectors among S1, . . . , Sn are linearly independent;

(S) Xi = −Xi in distribution.

Also, recall that in dimension d, the convex hull of a random walk, Cn, is almost surely a convexpolytope with a boundary ∂Cn equal to a union of faces F ∈ Fn such that almost surely, F isa (d-1)-dimensional simplex of the form:

F = Conv(Si1(F ), . . . , Sid(F )).

11.1 Face probabilities and expected number of faces

Theorem 8. Consider a d-dimensional random walk with i.i.d. increments and (H), (G)) and(S) are satisfied. Then

P (Conv(Si1 , . . . , Sid) ∈ Fn) = 2(2i1 − 1)!!

(2i1)!!

(2(n− id)− 1)!!

(2(n− id))!!

d−1∏k=1

1

ik+1 − ik,

for any d ≥ 1 and any choice of indices 0 ≤ i1 < · · · < id ≤ n.

Proof. Exercise: Use path decomposition and Sparre Andersen + Baxter.

This theorem leads to a number of corollaries, in particular:

n−id+i1∑i=0

P (Conv(Si, . . . , Si+id−i1) ∈ Fn) = 2d−1∏k=1

1

ik+1 − ik(38)

and

E[|Fn|] = 2∑

k1+···+kd−1≤nk1,...,kd−1≥1

1

k1 · k2 · · · kd−1

. (39)

26

11.2 Absorption probability in the planar case

A particularly interesting application of the previous theorem when d = 2 is the following

Theorem 9. Consider a planar random walk with i.i.d. increments and (H), (G)) and (S) aresatisfied. Then

P (0 /∈ Conv(S1, . . . , Sn)) =n∑k=1

(2(n− k)− 1)!!

k (2(n− k))!!.

Proof. The proof relies on a property peculiar to d = 2, namely the fact that

E[Card {F ∈ Fn | 0 is a vertex of F}] =

2× P (0 /∈ Conv(S1, . . . , Sn)) + 0× P (0 ∈ Conv(S1, . . . , Sn)).

It suffices then to apply the previous theorem, with i1 = 0 and S0 ≡ 0.

11.3 Expectation of other additive functionals of faces

More generally, if g is a non-negative, symmetric Borel function of (d−1)-dimensional simplicesin Rd, and if one writes

Gn =∑F∈Fn

g(Si2(F ) − Si1(F ), . . . , Sid(F ) − Sid−1(F )

),

then one has the following theorem.

Theorem 10. Let S1, . . . , Sn be the partial sums of n-exchangeable random vectors in Rd sat-isfying hypothesis (G).

E[Gn] = 2∑

1≤i−1<···<id−1≤n

g(Si1 , Si2 − Si1 , . . . , Sid−1

− Sid−2

)i1 (i2 − i1) · · · (id−1 − id−2)

.

In the cases when the increments are actually i.i.d., this leads further to:

E[Gn] = 2∑

k1+···+kd−1≤nk1,...,kd−1≥1

g(S

(1)k1, . . . , S

(d−1)kd−1

)k1 k2 · · · kd−1

,

where the S(l)k ’s are independent copies of the random walk.

Proof. Exercise.

11.4 Absorption probabilities for higher dimensional random walks

The argument used in Section 11.3 to compute the (non-)absorption probability of the origin inthe interior of the convex hull of the random walk was of course totally specific to dimension 2.In higher dimensions, it takes a different approach to be able to compute this probability. Itwas discovered a few years ago by Kabluchko, Vysotsky & Zaporozhets [25].

27

Theorem 11. Let S1, S2, . . . , Sn be a random walk in Rd with i.i.d., centrally symmetric in-crements X1, . . . , Xn & such that P (X1 ∈ H) = 0 for any affine hyperplane H (i.e. what wecalled earlier hypothesis (H) is satisfied). Then

P (0 /∈ Conv(S1, . . . , Sn)) =2

2n n!

dd/2e∑k=1

B(n, d− 2k + 1),

where the B(n, k)’s are the coefficients of the polynomial

(t+ 1)(t+ 3) . . . (t+ 2n− 1) =n∑k=0

B(n, k) tk.

Note that with d = 1, one recovers Sparre Andersen’s famous formula:

P (S1 > 0, . . . , Sn > 0) =(2n− 1)!!

(2n)!!,

so that this theorem is properly the multidimensional generalization of Sparre Andersen’s result.

Proof. The proof is based on a remarkable combinatorial identity leading to:

P (0 ∈ Conv(S1, . . . , Sn)) =Nn,d

2n n!,

whereNn,d is the number of Weyl chambers of type Bn intersected by any generic linear subspaceof Rn with co-dimension d. One then makes use of results from the theory of hyperplanearrangements to express this number as a sum of B(n, k) coefficients. As for the first part of theidentity, it relies on the fact that 0 ∈ Conv(S1, . . . , Sn) if and only if there exists a non-trivialnull linear combination of an s-permutation of X1, . . . , Sn with non-negative coefficients.

28

12 Multiple higher dimensional Gaussian random walks

In this section we present a more general and rigorous study of the “multi-paths” case discussedbriefly above for Brownian motions (see Section 8). More specifically, we show how one mayderive explicit formulae for the expected volume and the expected number of faces of theconvex hull of several multidimensional Gaussian random walks in terms of the (1-dimensional)Gaussian persistence probabilities. Special cases include the d-dimensional Gaussian polytopewith or without the origin.

12.1 Setting

Fix m,n1, . . . , nm ∈ N∗, and let

X(1)1 , . . . , X(1)

n1, . . . , X

(m)1 , . . . , X(m)

nm ,

be independent d-dimensional standard Gaussian vectors.Consider the collection of m random walks (S

(1)i )n1

i=1, . . . , (S(m)i )nmi=1 defined as

S(l)i = X

(l)1 + · · ·+X

(l)i , 1 ≤ l ≤ m, 1 ≤ i ≤ nl. (40)

We aim to study the properties of the joint convex hulls of the m random walks with andwithout the origin:

C0d := Conv

(0, S

(1)1 , . . . , S(1)

n1, . . . , S

(m)1 , . . . , S(m)

nm

)and

Cd := Conv(S

(1)1 , . . . , S(1)

n1, . . . , S

(m)1 , . . . , S(m)

nm

).

Some properties of Cd and C0d are convenient to formulate simultaneously. To this end, we

assume that C∗d might denote any of C0d , Cd.

With probability 1, C∗d is a convex polytope with boundary of the form

∂ C∗d =⋃

F∈F(C∗d)

F, (41)

where F(C∗d) stands for the set of (d−1)-dimensional faces of C∗d .Each face is a (d−1)-dimensional simplex almost surely.

Fix some non-negative integers

k1, . . . , km such that kl ≤ nl andm∑l=1

kl = d, (42)

and fix d indices

1 ≤ i(l)1 < · · · < i

(l)kl≤ nl for l = 1, . . . ,m such that kl > 0. (43)

Denote by Sd a d-tuple defined as

Sd :=

(S

(1)

i(1)1

, . . . , S(1)

i(1)k1

, . . . , S(m)

i(m)1

, . . . , S(m)

i(m)km

)(44)

29

with the convention that {S

(l)

i(l)1

, . . . , S(l)

i(l)kl

}:= ∅ for kl = 0,

and we also write

Conv Sd := Conv

{S

(1)

i(1)1

, . . . , S(1)

i(1)k1

, . . . , S(m)

i(m)1

, . . . , S(m)

i(m)km

}.

The same way fix some non-negative integers

k1, . . . , km such that kl ≤ nl andm∑l=1

kl = d− 1, (45)

and fix d− 1 indices

1 ≤ i(l)1 < · · · < i

(l)kl≤ nl for l = 1, . . . ,m such that kl > 0. (46)

Simiraly, let

S0d :=

(0, S

(1)

i(1)1

, . . . , S(1)

i(1)k1

, . . . , S(m)

i(m)1

, . . . , S(m)

i(m)km

)(47)

and

Conv S0d := Conv

{0, S

(1)

i(1)1

, . . . , S(1)

i(1)k1

, . . . , S(m)

i(m)1

, . . . , S(m)

i(m)km

}.

Note that Conv Sd may or may not be a face of Cd. Moreover, every face F ∈ F(Cd) canbe represented as Conv Sd with Sd defined in (44) with some integers (42) and indices (43).Therefore, we arrive at the following crucial albeit elementary relation: with probability one,∑

F∈F(Cd)

g(F ) =∑

k1+···+km=d0≤kl≤nl, l=1,...,m

∑1≤i(l)1 <···<i(l)kl ≤nll=1,...,m : kl>0

g(Sd)1{Sd∈F(Cd)}, (48)

where g : Rd → R1 is an arbitrary symmetric, non-negative, measurable function. Here, wewrite

g(F ) := g(xF1 , . . . ,xFd ), where xF1 , . . . ,x

Fd are the vertices of F .

For the convex hull with the origin, the appropriate relation is a little bit more complicated:with probability one,∑

F∈F(C0d)

g(F ) =∑

k1+···+km=d0≤kl≤nl, l=1,...,m

∑1≤i(l)1 <···<i(l)kl ≤nll=1,...,m : kl>0

g(Sd)1{Sd∈F(C0d)} (49)

+∑

k1+···+km=d−10≤kl≤nl, l=1,...,m

∑1≤i(l)1 <···<i(l)kl ≤nll=1,...,m : kl>0

g(S0d)1{Sd∈F(C0

d)}.

We aim to calculate the expectation of the right-hand side in (48) and (49), which by takingg ≡ 1 will readily give us the expected number of faces of Cd and Cd0. Moreover, by making amore subtle choice of g, we will be able to find the expected volumes of Cd and C0

d .

30

12.2 Notations

Our results will be expressed in terms of the unconditional and conditional Gaussian persistenceprobabilities: for r ∈ R1 let

pn(r) := P[ k∑i=1

Ni ≤ r, k = 1, . . . , n],

qn(r) := P[ k∑i=1

Ni ≤ r, k = 1, . . . , n∣∣ n∑i=1

Ni = r],

where N1, . . . , Nn are independent standard Gaussian random variables. Due to the symmetryof the distribution, for any r ≥ 0 the definition of qn(r) is equivalent to

qn(r) := P[ k∑i=1

Ni ≥ 0, k = 1, . . . , n∣∣ n∑i=1

Ni = r]. (50)

Note that

p1(r) = Φ(r), q1(r) =

{1, r ≥ 0,

0, r < 0,

and

pn(0) =(2n− 1)!!

(2n)!!, qn(0) =

1

n, (51)

where Φ(r) =∫ r−∞ ϕ(s) ds is the cumulative distribution of the standard Gaussian density ϕ(r),

and (51) was established by Sparre Andersen [?, ?] (who proved it in a setting much generalthan the Gaussian one considered here).

For a linear hyperplane L ⊂ Rd denote by PL the operator of the orthogonal projectiononto L. In particular, denote by Pd the operator of the orthogonal projection onto the firstd−1 coordinates:

Pd : x = (x1, . . . , xd−1, xd) 7→ (x1, . . . , xd−1, 0).

The d-dimensional volume is denoted by | · |. Some of the sets we consider have dimensiond−1, for them | · | stands for the (d−1)-dimensional volume. For finite sets, | · | denotescardinality.

Let Sd−1 ⊂ Rd denote the unit (d−1)-dimensional sphere centered at the origin and equippedwith the Lebesgue measure µ normalized to be probabilistic. For u ∈ Sd−1 denote by u⊥ thelinear hyperplane orthogonal to u. We write κd = πd/2/Γ(d

2+ 1) for the volume of the d-

dimensional unit ball.Let Q be a random orthogonal matrix chosen uniformly from the orthogonal group O(d)

with respect to the probabilistic Haar measure and independently with the random walks.

Remark 1. It is well known that the orthogonal projection of the standard Gaussian distributiononto a linear subspace is again the standard Gaussian distribution of corresponding dimension.Thus,

(PdS(1)i )n1

i=1, . . . , (PdS(m)i )nmi=1

can be considered as a collection of independent Gaussian random walks in Rd−1. In particular,we may think that

Pd Cd = Cd−1 and PdSd = Td−1, in law, (52)

where Td is defined similarly as Sd but with (d + 1) components instead of d. Thus Td is ad-simplex whereas Sd is a d-tuple of random points in Rd.

31

12.3 Expectation of a functional of a face

Theorem 12. Let g : Rd → R1 be some bounded measurable function invariant with respectto translations. Consider some Sd defined in (44) with some integers from (42) and indicesfrom (43). Then,

E[g(Sd)1{ConvSd∈F(Cd)}] =d!

2d/2+1Γ(d2

+ 1) E[g(PdSd) |convPdSd|

]×∏l : kl 6=0

[(i(l)1

(i(l)2 − i

(l)1

). . .(i(l)kl− i(l)kl−1

))−3/2

·(2(nl − i(l)kl )− 1)!!

(2(nl − i(l)kl ))!!

]

×∫ ∞−∞

exp

(− r2

2

∑l : kl 6=0

1

i(l)1

) ∏l : kl=0

pnl(r)∏l : kl 6=0

qi(l)1

(r) dr (53)

and

E[g(Sd)1{ConvSd∈F(C0d)}] =

d!

2d/2+1Γ(d2

+ 1) E[g(PdSd) |convPdSd|

]×∏l : kl 6=0

[(i(l)1

(i(l)2 − i

(l)1

). . .(i(l)kl− i(l)kl−1

))−3/2

·(2(nl − i(l)kl )− 1)!!

(2(nl − i(l)kl ))!!

]

×∫ ∞

0

exp

(− r2

2

∑l : kl 6=0

1

i(l)1

) ∏l : kl=0

pnl(r)∏l : kl 6=0

qi(l)1

(r) dr (54)

Also consider some S0d defined in (47) with some integers from (45) and indices from (46).

Then,

E[g(S0d)1{ConvSd∈F(C0

d)}] =

√2πd!

2d/2+1Γ(d2

+ 1) · E[g(PdS

0d) |convPdS

0d|]

×m∏l=1

[(i(l)1

(i(l)2 − i

(l)1

). . .(i(l)kl− i(l)kl−1

))−3/2

·(2(nl − i(l)kl )− 1)!!

(2(nl − i(l)kl ))!!

], (55)

with the convention that the factor in the product equals (2nl − 1)!!/(2nl)!! for kl = 0.

32

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