ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME

ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME Alexey Chuprunov Kazan State University, Russia István Fazekas University of Debrecen, Hungary 2012

Kolchin s generalized allocation scheme A law of large numbers

A law of large numbers Kolchin s generalized allocation scheme In the generalized scheme of allocations of particles into cells, the distribution of the cell contents are represented as the conditional distribution of independent random variables under the condition that their sum is fixed, see V. F. Kolchin (1999). Let η 1,..., η N be nonnegative integer-valued random variables. If there exist independent random variables ξ 1,..., ξ N such that P{η 1 = k 1,..., η N = k N} = (1.1) = P{ξ 1 = k 1,..., ξ N = k N ξ 1 + + ξ N = n}, where k 1,..., k N are arbitrary nonnegative integers, we say that η 1,..., η N represent a generalized allocation scheme with parameters n and N, and independent random variables ξ 1,..., ξ N.

A law of large numbers Let the random variables ξ 1,..., ξ N be identically distributed. Usually (see, e.g., A. V. Kolchin (2003)) the random variables ξ 1,..., ξ N are distributed as follows q k = P{ξ 1 = k} = (b k θ k )/(k!b(θ)) where b 0, b 1,... is a certain sequence of non-negative numbers, and B(θ) = b k θ k /k!. k=0

Example. Poisson polynomial A law of large numbers Let ξ i have Poisson distribution, i.e. P(ξ i = k) = λk k! e λ, k = 0, 1,.... Then ( ) n! 1 n P{ξ 1 = k 1,..., ξ N = k N ξ 1 + + ξ N = n} = k 1!... k N! N if k 1 + + k N = n. That is {η 1 = k 1,..., η N = k N} has polynomial distribution. More precisely, it describes the allocation of n balls into N boxes. That is we obtain the original allocation scheme.

A law of large numbers Example. Binomial poly-hypergeometric Let ξ i have binomial distribution, i.e. P(ξ i = k) = ( t k) p k (1 p) t k, k = 0, 1,..., t. Then ( t P{ξ 1 = k 1,..., ξ N = k N ξ 1 + +ξ N = n} = k 1 ) ( t... k N ) / ( ) Nt n if k 1 + + k N = n. That is {η 1 = k 1,..., η N = k N} has poly-hypergeometric distribution.

Example. Random forest A law of large numbers Let T n,n denote the set of forests containing N labelled roots and n labelled nonroot vertices. By Cayley s theorem, T n,n has N(n + N) n 1 elements. Consider uniform distribution on T n,n. Let η i denote the number of the nonroot vertices of the ith tree. Then P{η 1 = k 1,..., η N = k N} = n! (k 1 + 1) k1 1... (k N + 1) k N 1 k 1!... k N! N(N + n) n 1. Now let ξ i have the following distribution (Borel, Tanner) P(ξ i = k) = λk (1+k) k 1 k! e (k+1)λ, k = 0, 1,..., λ > 0. Then P{ξ 1 = k 1,..., ξ N = k N ξ 1 + +ξ N = n} = P{η 1 = k 1,..., η N = k N}. See Kolchin (1999), Ch-F (2010).

A law of large numbers Consider a sequence of non-negative numbers b 0, b 1,... with b 0 > 0, b 1 > 0 (1.2) and assume that the convergence radius R of the series B(θ) = k=0 b kθ k /k! (1.3) is positive. Let us introduce the random variable ξ = ξ(θ) (where θ > 0) with distribution p k (θ) = P{ξ = k} = b k θ k /k!b(θ), k = 0, 1, 2,.... (1.4) By Kolchin (2003), one has m = m(θ) = Eξ = θb (θ)/b(θ). Let 0 < θ < θ < R. m(θ), θ [θ, θ ], is a positive, continuous, strictly increasing function. We will denote by m 1 the inverse function of m.

A law of large numbers A law of large numbers In view of independence of the random variables ξ 1,..., ξ N, the study of several questions of the generalized allocation scheme can be reduced to problems of sums of independent random variables. Let µ nn be the number of the random variables η 1,..., η N being equal to a fixed r, r = 0, 1,..., n. Observe that µ nn = µ rnn = N i=1 I {η i =r} (1.5) is the number of cells containing r particles.

A law of large numbers Theorem B (Ch-F 2010) Suppose that ξ 1 = ξ 1 (θ) has distribution (1.4) with property (1.2). Assume that n, N such that n N = α nn α where 0 < α <. Assume that 0 < θ < R where θ = m 1 (α). Then we have lim n,n 1 N µ nn = p r (θ) almost surely.

In this paper we study the following scheme. Let ξ 0, ξ 1,..., ξ N be independent identically distributed random variables such that P{η 1 = k 1,..., η N = k N } = (2.1) = P{ξ 1 = k 1,..., ξ N = k N ξ 1 + + ξ N n}, where k 1,..., k N are arbitrary nonnegative integers. Then we say that η 1,..., η N obey a generalized allocation scheme. It describes the generalized allocation of at most n balls into N boxes. Let p r = P(ξ 0 = r), r N 0. a = Eξ 0 <.

Observe that µ nn = µ rnn = N I {η i=1 i =r} (2.2) is the number of cells containing r particles. Introduce notation S N = N i=1 ξ i, S c N = N i=1 (ξ i a). Throughout the paper Φ denotes the distribution function of the standard Gaussian law. Notation α nn = n N.

Theorem Let Eξ 0 = a <. Suppose that there exists a sequence B N, N = 1, 2,..., such that N B N > 0 for all N = 1, 2,..., B N, B N 1 lim = 1, (2.1) N B N and 1 B N S c N ξ, as N, (2.2) in distribution where ξ is an almost surely finite random variable. Denote by F the distribution function of ξ. Let C be a point of continuity of F such that F (C) = P{ξ < C} = q 0 > 0. Let C < d 1 <. Then we have 1 lim n,n, a+c B N N <α nn <d 1 N µ nn = p r almost surely. (2.3)

Corollary Let Eξ 0 = a <. Let a < d < d 1 <. Then we have 1 lim n,n, d<α nn <d 1 N µ nn = p r almost surely. Corollary Suppose that Eξ 2 0 < and let Eξ 0 = a. Let C > 0, a < d 1 <. Then we have 1 lim n,n, a C <α N nn <d 1 N µ nn = p r almost surely.

Main results of this subsection are contained in the following theorem. We study the convergence of 1 N µ nn under the condition: n, N such that n N = α nn α. We show that an analogue of Kolmogorov s SLLN is valid if α is large and an analogue of Theorem B is valid if α is small. Moreover, 1 N µ nn is convergent, if ξ 0 satisfies Cramér s condition. Recall that ξ satisfies Cramér s condition, if there exists a positive constant H such that Ee λξ < for all λ < H. Observe that a positive random variable ξ satisfies Cramér s condition, if and only if Ee λξ < for some 0 < λ. We emphasize that next Theorem concerns the general model (2.1), and the particular distribution (1.4) is assumed only in part (3) of the theorem.

Theorem Let Eξ 0 = a <. (1) Let α > a. Then we have 1 lim n,n, α nn α N µ nn = p r almost surely. (2) Assume that ξ 0 satisfies the Cramér condition. Then we have 1 lim n,n, α nn a N µ nn = p r almost surely. (2.4) (3) Let 0 < α < a. Suppose that the random variable ξ 0 = ξ 0 (θ) has distribution (1.4) and condition (1.2) is valid. Assume that θ = m 1 (α) and 0 < θ < R. Then we have 1 lim n,n, α nn α N µ nn = p r (θ) in probability. (2.5)

Denote by ξ (r) 0 a random variable with distribution P{ξ (r) 0 = k} = P{ξ 0 = k ξ 0 r}. Remark that P(ξ 0 r) 0, because ξ 0 is nondegenerate. Let Eξ (r) = a r and let ξ (r) 1,..., ξ(r) N be independent copies of ξ(r) 0. Let S (r) N = N i=1 ξ(r) i. CN k = ( ) N k is the binomial coefficient. We have the following analogue of Kolchin s (1999) formula for model (2.1). Theorem For all k = 0, 1, 2,..., N we have (r) P{µ nn = k} = CN k P{S pk N k N k r (1 p r ) n kr}. (2.6) P{S N n}

Formula (2.6) gives us possibility to apply usual limit theorems for sums of independent random variables in order to prove local limit theorems for µ nn. Theorem (1) Let d > a. Let s 2 r = p r (1 p r ). Then, uniformly for α nn > d, we have P{µ nn = k} = 1 2πNsr e u2 /2 (1 + o(1)), (2.7) as n, N and u = k Npr belongs to an arbitrary bounded fixed s r N 1/2 interval. (2) Suppose a < a r and a < d < d 1 < a r. Then, for any fixed k, we have lim P{µ nn = k} = 0. (2.8) n,n, d<α nn <d 1

Theorem Suppose that Eξ0 2 <. Denote by σ2 the variance of ξ 0 and by σr 2 the variance of ξ (r) 0. Let < C. Then, as n, N such that N(α nn a) C, we have = P{µ nn = k} = (2.9) (( ( ) 1 e u2 /2 C r a usr / ( ) ) 1 p Φ 1 r C Φ + o(1)) 2πNsr pr σ r σ for u = k Npr s r N 1/2 belonging to any bounded fixed interval.

Theorem Suppose that the random variable ξ 0 = ξ 0 (θ) has distribution (1.4), condition (1.2) is satisfied, and θ K < R. Suppose that r = 1 and n k is fixed. Let N such that Np r (θ) λ for some 0 < λ <. Then for all k N 0 we have P{µ nn = k} = λ k e λ k! n (1 + o(1)). (2.10) l=0 λl e λ l! Corollary Suppose that the random variable ξ = ξ(θ) has distribution (1.4), condition (1.2) is satisfied, and θ K < R. Suppose that r = 1. Let n, N such that Np r (θ) λ for some 0 < λ <. Then for all k N 0 we have P{µ nn = k} = λk e λ k! (1 + o(1)).

The following theorem shows that, using stronger conditions, the previous Corollary is valid for r > 1. Theorem Suppose that the random variable ξ = ξ(θ) has distribution (1.4), condition (1.2) is satisfied, and θ K < R. Let r > 1 and n N 1 1/r. Let N such that Np r (θ) λ for some 0 < λ <. Then for all k N 0 we have P{µ nn = k} = λk e λ (1 + o(1)). (2.11) k! Previous Theorems are analogues of certain results in Kolchin (1999).

Let η (N) = max 1 i N η i. Now let ξ ( r) 0 be a random variable with distribution P{ξ ( r) 0 = k} = P{ξ 0 = k ξ 0 r}. Let ξ ( r) i, i = 1,..., N, be independent copies of ξ ( r) 0. Let S ( r) N = N i=1 ξ( r) i, a r = Eξ ( r) 0, P r = P{ξ 0 > r}. We have the following analogue of Kolchin s (1999) formula. Theorem ( r) N P{S N n} P{η (N) r} = (1 P r ), for all r N, (2.12) P{S N n} P{η (N) 0} = (1 P 0) N P{S N n}.

By (2.12), we can apply limit theorems for sums of i.i.d. r.v. s. Theorem Let d > a. Then for all r N, as n, N, we have uniformly for α nn > d. P{η (N) r} = (1 P r ) N (1 + o(1)) (2.13) Theorem Suppose that Eξ0 2 <. Let < C. Then, for all r N, as n, N such that N(α nn a) C we have / ( )) ) C P{η (N) r} = (1 P r ) ((1 N Φ + o(1), σ for a r < a, (2.14) P{η (N) r} = (1 P r ) N ((1 + o(1))), a r = a. (2.15)

Theorem Suppose that the random variable ξ = ξ(θ) has distribution (1.4), condition (1.2) is satisfied, and θ K < R. Let r N and b r 0. Let θ = θ(n) be such that Np r+1 (θ) λ where 0 < λ <. Then, as n, N such that n N r/(r+1), we have P{η (N) = r} = e λ + o(1), (2.16) P{η (N) = r + 1} = 1 e λ + o(1). (2.17)

The previous theorem is valid with weaker conditions in the case of r = 0. Corollary Suppose that the random variable ξ = ξ(θ) has distribution (1.4), condition (1.2) is satisfied, and θ K < R. Let r = 0. Let θ = θ(n) be such that Np 1 (θ) λ where 0 < λ <. Then, as n, N, we have P{η (N) = 0} = e λ + o(1) and P{η (N) = 1} = 1 e λ + o(1). To prove it we have to use the next theorem.

Theorem Suppose that the random variable ξ = ξ(θ) has distribution (1.4), condition (1.2) is satisfied, and θ K < R. Let r = 0 and let n N be fixed. Let θ = θ(n) be such that Np 1 (θ) λ where 0 < λ <. Then, as N, we have P{η (N) = 0} = P{η (N) = 1} = 1 e λ n l=0 λl e λ l! e λ n l=0 λl e λ l! + o(1), (2.18) + o(1). (2.19)

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