Statistics & Probability Letters 8 (1989) 123-127 North-Holland

June 1989

POISSON LIMITS FOR GENERALIZED RANDOM ALLOCATION PROBLEMS

Bernard HARRIS

Department of Statistics, University of Wisconsin, Madison, WI 5371 I, USA

Received October 1987 Revised June 1988

Abstract: n balls are randomly distributed into N cells so that no cell may contain more than one ball. This process is repeated m times. In addition, balls may disappear, such disappearances are independent and identically Bernoulli distributed.

Conditions are given under which the number of empty cells has an asymptotically (N * cc) Poisson distribution.

AMS (MOS) Subject Chkjications: 6OCO6, 05A15, 60F05.

Keywords: random allocations, occupancy, Poisson limits.

1. Introduction

Let n, N be positive integers with n < N. Assume that n balls are randomly distributed into N cells, so that no cell may contain more than one ball. Then the probability that each of n specified cells will be occupied is (,“)-I. This process is repeated m times, so that there are (f)” random allo- cations of nm balls among the N cells. In ad- dition, for each ball, let p, 0 <p G 1, be the prob- ability that the ball will not “disappear” from the cell. The “disappearances” are assumed to be sto- chastically independent for each ball; thus the disappearances constitute a sequence of nm

Bernoulli trials.

Such random allocation processes may be viewed as filing or storage processes. Objects are randomly assigned to files or storage bins. From time to time, objects may be missing, that is, they have disappeared.

Several special cases of this problem have been considered previously. In particular p = 1, n = 1 is called the classical occupancy problem or the Maxwell-Boltzman model in statistical mecha- nics, see [3,4,10]. The case p = 1, n arbitrary is treated in [5,6,9,11]. The case m = 1 is known as the Fern-t-Dirac model in statistical mechanics. Poisson limits for such problems are given in [7,8]. If 0 <p < 1, then this is a randomized version of the above mentioned models.

In this paper the investigation begun in [2] is continued. There the probability distribution and moments of the number of empty cells were ob- tained. It was also shown that the number of empty cells has a representation as a sum of independent Bernoulli random variables and this fact was exploited in obtaining sufficient condi- tions for the number of empty cells to be asymp- totically (N + co) normally distributed. We now establish sufficient conditions for the number of empty cells to have an asymptotically Poisson distribution.

In Section 2, we summarize those results in [2], which will be used here. The sufficient conditions for an asymptotic Poisson distribution will be given in Section 3.

2. ‘The probability distribution and the moments of the number of empty cells

Sponsored by the United States Army under Contract No. Let m, n, N be positive integers with n <N. m

DAAG29-80-C-0041. sets of balls, consisting of n balls each, are distrib-

0167-7152/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland) 123

Volume 8, Number 2 STATISTICS & PROBABILITY LETTERS June 1989

uted into N cells at random so that no cell can contain more than one ball from the same set. Thus, at the end of this distribution, each cell has ui balls,

U, = 0, 1 ,...,m, i=1,2 ,..., N,

and E u,=mn. i=l

In addition, each ball may disappear with com- mon probability 1 - p, 0 < p < 1 and these disap- pearances are stochastically independent, con- stituting a sequence of mn Bernoulli trials. Let S

be the number of empty cells. Let P,,,,,,p(j) = P{S =j} and let E(S(“)) be the vth factorial moment of S. Let &(t) be the factorial moment generating function of S. Then, from [2], we have

E( S”‘)

= (;)-mN+o(l -P)‘( :,i($ and

E(S) = N(l -pn/N)m.

Further

$&> = IE (p(;j-” r=O

xjJgoo-P)J(~qq;)

(2)

m

1 . (3 1

The following lemma will also be used in the sequel.

Lemma 1.

Proof. Expand the right hand side in a binomial series obtaining

Thus, the coefficient of pJ is

(-1)‘i (;j( :I:)( :)I( :j. v=j

It can easily be seen (for example [l, p. 1051) that

from which the conclusion follows readily. q

3. Poisson limits in random allocations

In order to establish the conditions for a Poisson limit, the following preliminary lemmas will be employed.

Lemma2. For O<j<n<N,

Also, for fixed j, as n + CQ,

(5)

(6)

Proof. For j = 0, (r)/(r) = 1 G (n/N)j. Assume that forO(k<n,

Then

jk:l)/(kE:l)

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Volume 8, Number 2 STATISTICS & PROBABILITY LE’lTERS June 1989

Also, since e-X/(1-X) < 1 - x < eex, whenever 0 < since NC”) = N”(l + 0( N-l)) and n < N. The con- x < 1, we have clusion now follows readily. q

Lemma 4. Let m, n, N be positive integers with

n<Nandletpsatisfy O<p<l. Let N-+co and

m + cc so that

Similarly,

=-_e ;( -A- 1)/(2(-l)) 3

establishing the lemma. q

Lemma 3. Let m, n, N be positive integers with

n < N and let p satisfy 0 < p < 1. If for every fixed

m,

N(l -pn/N)m -+A>0 asN-+co, (7)

then

E(P) + 2

for every fixed non-negative integer v.

Proof. From Lemma 1,

E( S”‘)

= (;)-mN@“[;O(l-~)‘( ::;)(;)lm

=N”‘,j,r,(;)(;)(-l)‘~‘/(y)j~.

If (7) holds, then, for fixed m, n + co as N + co. Hence from Lemma 2,

E(S”‘) = N@)( ~O(;)(;)i(-l)’

i

m

Xpj(1 + O(n-l))

= N(“)[(l -pn/N)Ym(l + O(n-I))]

= N”(1 -pn/N)““‘(l + O(n-I)),

m/log N-+0 and N(l-np/N)m+X>O.

Then

E(S”‘) + A”

for every fixed non-negative integer v.

Proof. The hypotheses imply that

(1 -pn/N)m=X/N+o(NP1) as N-t cc.

Thus

(8)

(1 - pn/N) = (h/N + o( N-l))“m

= (h/N)““(l + o(1))““. (9)

Further, from (9) it is clear that as m + 00,

N+CQ,

m-l log( X/N) + - cc

and thus n - 00 as N + 00 and (6) can be ap- plied. Thus, we get

E( S’“‘)

= N”(l -pn/N)Ym(l + O(nP1))m+l,

since n < N. Since m/log N + 0 as N += co, the conclusion

follows. 0

Lemma 5. Let m( n, N), n, N be positive integers

withn<Nandletpsatisfy O<p<l. LetN+oo

and m -+ cc so that

m/log N -+ r > 0 asN+co.

Then if

N(l -rip/N))) + A > 0,

we have

E( S’“‘) + A”.

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June 1989 Volume 8, Number 2 STATISTICS ‘5% PROBABILITY LETTERS

Proof. From the hypotheses: Then, it can easily be seen that

log(1 - rip/N)) = m -1[1og( h/N) + log(1 + o(l))]. E( 9”‘) = N’“‘(1 - Wzp/N + o( (rip/N))))““““›

Since m+ co as N-+ 00,

m-’ log(A/N) + --r-l.

Thus,

0~ lim (np/N)<l, N-CC

and

n=aN+o(N), O<cy<l, as N-co.

Thus, n+ m as N-+ cc and

E(S”‘) = N’(1 -JVI/‘N)‘~(~ + O(n-‘)jm+r

as in the proof of Lemma 4. The conclusion follows, since

m/log N --j r > 0

implies

(1 + O(n-l))m = (1 + O(e(m/‘og N)(‘“g N/N))). 0

Lemma 6. Let m, n, N be positive integers with n<Nandletpsatisfy O<p<l. LetN+oo and

m -+ co so that

m/log N+ co.

Then if

N(l-np/N)m+h>O,

we have

E(s”‘) + A”.

Proof. From the hypotheses,

log(l - rip/N )

= m-‘[log( X/N) + log(1 + o(I))]

and m + co as N -+ co implies

m-’ log( h/N) + 0.

Thus,

rip/N --+ 0 as N+co.

I26

and consequently,

E(S(‘)) = N’ e- u”pm/N( 1 + 0( (rip/N))))

x(1 + O(N-‘)).

Thus, since

N(1 -pn/N)m

= N eemP”‘N(l + 0( m( pn/N)*))

as N + co, the conclusion follows. 0

Combining Lemmas 3, 4, 5, and 6, we have the following theorem.

Theorem. Let m(N), n(N), N be sequences of

positive integers with n < N and letp = p( N) satisfy 0 < p < 1. Then if

S, the number of empty cells in the random allo-

cation model has asymptotically (N + CXI) a Pois-

son distribution with mean A, and for every non-

negative integer v,

E( S’“‘) -+ A”. 0

References

[l] Harris, B. (1966) Theory of Probability (Addison-Wesley Publishing Company, Reading, MA).

[2] Harris, B., M. Marden and C.J. Park (1987), The distribu- tion of the number of empty cells in a generalized random allocation scheme, Ann. Discrete Math. 33, 77-90.

[3] Harris, B. and C.J. Park (1971). A note on the asymptotic normality of the distribution of the number of empty cells in occupancy problems, Ann. Inst. Statist. Math. 23, 507-513.

[4] Holst L. (1977), Some asymptotic results for occupancy problems, Ann. Probab. 5, 1028-1035.

[5] Holst, L. (1980) On matrix occupancy, committee, and capture-recapture problems, Stand J. Statist. 7, 139-146.

[6] Kolchin, V.F., B.A. Sevast’yanov and V.P. Chistyakov (1978) Random AlIocations (V.H. Winston and Sons, Washington, DC).

[7] Mikhailov, V.G. (1977), A Poisson limit theorem in the scheme of group disposal of particles, Theory Probab. Appl. 22, 152-156.

Volume 8, Number 2 STATISTICS & PROBABILITY LETTERS June 1989

[8] Mikhailov, V.G. (1977), An estimate of the rate of conver- gence to the Poisson distribution in group allocation of particles, Theory Probab. Appl. 22, 554-562.

[9] Park, C.J. (1981), On the distribution of the number of unobsenred elements when m-samples of ut ~1 xe drawn from a finite population, Comm. Statisr A Theory

Methods 10, 371-383.

[lo] Sevast’yanov, B.A. and V.P. Chistyakov (1964), Asymp- totic normality in the classical ball problem, Theory

Probab. Appl. 9, 198-211. [ll] Vat&in, V.A. and V.G. Mikhailov (1982), Limit theorems

for the number of empty cells in an equiprobable scheme for group allocation of particles. Theory Probab. Appl. 27,

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