A generalized allocation scheme

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1 Annales Mathematicae et Informaticae 39 (202) pp Proceedings of the Conference on Stochastic Models and their Applications Faculty of Informatics, University of Debrecen, Debrecen, Hungary, August 22 24, 20 A generalized allocation scheme István Fazekas a, Bettina Porvázsnyik b a University of Debrecen, Faculty of Informatics Debrecen, Hungary fazekasi@inf.unideb.hu b University of Debrecen, Faculty of Science and Technology Debrecen, Hungary porv.bettina@gmail.com Dedicated to Mátyás Arató on his eightieth birthday Abstract The generalized allocation scheme was introduced by V.F. Kolchin []. Let ξ, ξ 2,..., ξ be independent identically distributed non-negative integer valued non-degenerate random variables. Consider the random variables η,..., η with joint distribution Pη = k,..., η = k = P ξ = k,..., ξ = k ξi = n. Let ξ i have Poisson distribution, then (η,..., η ) has polynomial distribution. Therefore η = k,..., η = k means that the contents of the boxes are k,..., k after allocating n balls into boxes during the usual allocation procedure. Our aim is to study random variables η,..., η with joint distribution Pη = k,..., η = k = P ξ = k,..., ξ = k ξi n. It can be considered as a general allocation scheme when we place at least n balls into boxes. Let µ n denote the number of cases when η i = r. That is µ n is the number of boxes containing r balls. We shall prove limit theorems for Pµ n = k. Moreover, we shall consider the asymptotic behaviour of Pmax i η i r and Pmin i η i r. Keywords: generalized allocation scheme, conditional probability, law of large numbers, central limit theorem, Poisson distribution. MSC: 60C05, 60F05 Supported by the Hungarian Scientific Research Fund under Grant o. OTKA T07928/2009. Supported by the TÁMOP-4.2.2/B-0/ project. The project is co-financed by the European Union and the European Social Fund. 57

2 58 I. Fazekas, B. Porvázsnyik. Introduction The usual allocation scheme is the following. Let n balls be placed successively and independently into boxes. At any allocation the ball can fall into each box with probability /. This model was widely studied. See the early papers Weiss [3], Rényi [2], Békéssy [] and the monograph Kolchin-Sevast yanov-chistyakov [8]. See also Chuprunov-Fazekas [2] for certain recent results. A generalization of the usual allocation scheme was introduced by V.F. Kolchin (see the monographs of Kolchin [7] and Pavlov [0]). Let η, η 2,..., η be nonnegative integer-valued random variables. In Kolchin s generalized allocation scheme the joint distribution of η, η 2,..., η can be represented as Pη = k,..., η = k = P ξ = k,..., ξ = k ξ i = n, (.) where ξ, ξ 2,..., ξ are independent identically distributed non-negative integer valued non-degenerate random variables and k, k 2,..., k are arbitrary non-negative integers, k + k k = n. This scheme contains the usual allocation procedure, certain random forests, and several other models (see the monographs of Kolchin [7] and Pavlov [0]). The usual allocation scheme is obtained as follows. Let ξ i have Poisson distribution, i.e. P(ξ i = k) = λk k! e λ, k = 0,,.... Then ( ) n n! Pξ = k,..., ξ = k ξ + + ξ = n = k!... k! if k + + k = n. That is (η,..., η ) has polynomial distribution. ow η = k,..., η = k means that the cell contents are k,..., k after allocating n particles into cells considering the usual allocation procedure. The connection of the random forest and the generalized allocation scheme is the following. Let T n, denote the set of forests containing labelled roots and n labelled non-root vertices. By Cayley s theorem, T n, has (n + ) n elements. Consider uniform distribution on T n,. Let η i denote the number of the non-root vertices of the ith tree. Then Pη = k,..., η = k = n! (k + ) k... (k + ) k k!... k! ( + n) n. ow let ξ i have Borel distribution (see [5], [9]) P(ξ i = k) = λk (+k) k k! e (k+)λ, k = 0,,..., λ > 0. Then Pξ = k,..., ξ = k ξ + + ξ = n = n! (k + ) k... (k + ) k k!... k! ( + n) n if k + + k = n. See [7, 2, 0]. Therefore η,..., η satisfy (.).

3 A generalized allocation scheme 59 We can say that in the generalized allocation scheme we place n balls into boxes. In the framework of the generalized allocation scheme several asymptotic results can be obtained. Let µ r be the number of the random variables η, η 2,..., η being equal to r (r = 0,,..., n). Observe that µ r = µ rn = µ n = I η i =r (.2) can be considered as the number of boxes containing r balls. Here I A is the indicator of the set A, i.e. I A (x) = if x A and I A (x) = 0 if x A. (µ r, µ rn, and µ n are just different notations for the same quantity.) Limit results for µ r can be obtained in the following way. Let ξ 0 be a random variable with the same distribution as ξ. Let p r = Pξ 0 = r and Eξ 0 = a. Introduce notation S = ξ i. Denote by ξ (r) 0 a random variable with distribution Pξ (r) 0 = k = Pξ 0 = k ξ 0 r. (.3) The expectation and the second moment of ξ (r) 0 are the following a r = Eξ (r) 0 = a rp ( ) 2 r and E ξ (r) Eξ 0 = 0 2 r 2 p r. Let ξ (r) p r p,..., ξ(r) be independent copies of r ξ (r) 0. Let S(r) = ξ(r) i. Denote by C k the binomial coefficient Ck = ( ) k. V.F. Kolchin proved in [7] the following lemma. Lemma.. Let µ n and ξ(r) 0 be defined by (.2) and (.3), respectively. Then Pµ n = k = Cp k k PS(r) k k r( p r ) = n kr. (.4) PS = n Using this representation, normal and Poisson limit theorems were obtained (see [7], and [0]). In [4] a modification of the generalized allocation scheme was studied, that is in (.) the condition was changed for ξ i n. In this paper we introduce another scheme, i.e. we use in (.) condition of the form ξ i n. It can be considered as a general allocation scheme when we place at least n balls into boxes. Let µ n denote the number of cases when η i = r. That is µ n is the number of boxes containing r balls. We shall prove limit theorems for Pµ n = k. Moreover, we shall consider the asymptotic behaviour of Pmax i η i r and Pmin i η i r. In Section 2 Pµ n = k is studied. In sections 3 and 4 Pmax i η i r and Pmin i η i r are considered, respectively. 2. Another generalized allocation scheme Let ξ, ξ 2,..., ξ be independent identically distributed non-negative integer-valued non-degenerate random variables. Consider random variables η, η 2,..., η with

4 60 I. Fazekas, B. Porvázsnyik joint distribution Pη = k,..., η = k = P ξ = k,..., ξ = k ξ i n. (2.) In this case, we place at least n balls into boxes. Example 2.. Let ξ i have Poisson distribution, i.e. 0,,.... Then Pξ = k,..., ξ = k ξ + + ξ n = P(ξ i = k) = λk k! e λ, k = / k!... k! λk0 k=n (λ) k k! (2.2) if k + + k = k 0 n. ow, we place η (random number) balls into boxes. Assume that η n. Let η i denote the number of balls in the ith box. Then Pη = k,..., η = k = = P(η = k,..., η = k η = i)p(η = i) i=n k 0! k!... k! ( ) k0 P(η = k 0 ), (2.3) if k + + k = k 0 n. If we choose the a priori distribution of η as Poisson distribution truncated from below, i.e P(η = i) = (λ)i e λ/ i! k=n (λ) k e λ, i = n, n +,..., k! then we obtain (2.2). That is our scheme (2.) with ξ i having Poisson distribution describes the usual allocation when the number of balls is given by a truncated Poisson distribution. Let µ r = µ rn = µ n = I η i=r be the number of the boxes containing r balls. Then we have the following analogue of Kolchin s formula (.4) for our model. Recall that ξ (r) 0 is defined by (.3). Theorem 2.2. For all k = 0,, 2,..., we have Pµ n = k = Cp k k PS(r) k k r( p r ) n kr. (2.4) PS n Proof. (2.4) can be proved by a certain modification of the proof of Lemma.. Let A (r) k be the event that exactly k of the random variables ξ,..., ξ are equal to r. By (2.), we have Pµ n = k = P(A (r) k S n) = P(A(r) k, S n). P(S n)

5 A generalized allocation scheme 6 Furthermore, P(A (r) k, S n) = P(S n A (r) k )P(A(r) k ) = C k p k r( p r ) k P(S n ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) = C k p k r( p r ) k P(S (r) k n kr). Here we have used that ξ,..., ξ are independent random variables and the event can occur C k different ways, moreover A (r) k P(S n A (r) k ) = P(S n, A (r) k ) P(A (r) k ) = P(ξ + + ξ n, ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) P(ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) = P(ξ + + ξ k n kr, ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) P(ξ r,..., ξ k r, ξ k+ = r,..., ξ = r) = P(ξ + + ξ k n kr, ξ r,..., ξ k r). P(ξ r,..., ξ k r) The proofs of our limit theorems are based on representation (2.4). First we consider two theorems with normal limiting distribution. Let α n = n. Theorem 2.3. Let Eξ 0 = a be finite, Eξ (r) 0 = a r, s 2 r = p r ( p r ). () Let d < a. Then, uniformly for α n < d, we have Pµ n = k = 2πsr e u2 /2 ( + o()), (2.5) as n, and u = k pr s r belongs to an arbitrary bounded fixed interval. /2 (2) Suppose that a r < a. Let a r < d < d < a. If k belongs to a bounded interval, then we have lim Pµ n = k = 0. (2.6) n,, d <α n <d Proof. () By the Moivre-Laplace Theorem we have C k p k r( p r ) k = as uniformly if u = k pr s r /2 s 2 r = p r ( p r ). lim P n,, α n <d 2πsr e u2 /2 ( + o()), (2.7) belongs to a bounded fixed interval, where As α n < d < a, applying Kolmogorov s law of large numbers, we obtain k lim P n,, α n <d ξ (r) i n kr ξ i n =. =, (2.8)

6 62 I. Fazekas, B. Porvázsnyik ow (2.4), (2.7) and (2.8) imply (2.5). (2) Let d < α n < d. By Kolmogorov s law of large numbers, we have lim P n,, d <α n <d k ξ (r) i n kr We obtain (2.6) from (2.4), if we apply (2.7) and (2.9). = 0. (2.9) Remark 2.4. It is easy to see that a < a r, a > a r and a = a r if and only if a > r, a < r and a = r, respectively. Let Φ denote the standard normal distribution function. Recall that a = Eξ 0, a r = Eξ (r) 0 and s 2 r = p r ( p r ). Theorem 2.5. Suppose that Eξ0 2 <. Denote by σ 2 the variance of ξ 0 and by σr 2 the variance of ξ (r) 0. Assume 0 < σ2, σr 2 <. Let C <. Then, as n, such that (α n a) C, we have ( ) a r C+us r Φ pr prσ r Pµ n = k = e u2 /2 2πsr Φ ( ) + o() C, (2.0) σ for u = k pr s r /2 belonging to any bounded fixed interval. Proof. As σ 2 = D 2 (ξ 0 ) < and σr 2 = D 2 (ξ (r) 0 ) <, by the central limit theorem, we obtain P ξ i n = P ξ i a (αn a) σ σ ( ) (αn a) = Φ + o(), (2.) σ and similarly we obtain P k ξ (r) i n kr = Φ a r (αn a) + us r p r + o(). (2.2) pr σ r Using (2.), (2.2), and (2.7), relation (2.4) implies the desired result. Using large deviation theorems we can describe the relation between µ n and µ n. Let X, X 2,... be independent identically distributed non-negative non-degenerate random variables with lattice distribution (assume that the span of the

7 A generalized allocation scheme 63 distribution of X is ). Suppose that Cramér s condition is satisfied, that is Ee λ0x < for some λ 0 > 0. Let Z = X + + X. Introduce notation M(h) = Ee hx, a(h) = (ln(m(h))), v 2 (h) = a (h). As X is non-degenerate, therefore a (h) > 0, so a( ) is strictly increasing. We have the following lemma from []. Lemma 2.6. Let x be an integer number and let h = a ( x ). Then, as, we have P(Z = x) = P(Z x) = ( ( v(h) 2π M (h)e hx + O v(h) 2π M (h)e hx ( e h ) )), ( ( )) + O uniformly for x, with a(ε) x a(λ 0 ε), where ε is an arbitrary small positive number. In particular Introduce notation P(Z x) P(Z = x) = ( e h ) ( + o()). (2.3) L(λ) = Ee λξ0, L r (λ) = Ee λξ(r) 0 where we assume that there exist positive constants λ 0 > 0 and λ (r) 0 > 0 such that Ee λ0ξ0 < and Ee λ(r) 0 ξ(r) 0 < (Cramér s condition). Let m(λ) = (ln(l(λ))), σ 2 (λ) = m (λ), 0 λ λ 0, m r (λ) = (ln(l r (λ))), σ 2 r(λ) = m r(λ), 0 λ λ (r) 0. As ξ 0 is non-degenerate, therefore m(.) is strictly increasing. Assume that 0 < P(ξ 0 = 0) <. Moreover, if we additionally assume that r 0 and P(ξ 0 = 0) + P(ξ 0 = r) <, then ξ (r) 0 is non-degenerate, therefore similar property is valid for the function m r (.). Let h = m (α n ), h r = m r (α n ), and β(α n ) = e h e. hr Theorem 2.7. Assume r > 0, P(ξ 0 = 0) > 0, and P(ξ 0 = 0) + P(ξ 0 = r) <. Let maxa, a r < d < d 2 < minm(λ 0 ), m r (λ (r) 0 ). Then, as n,, we have uniformly for d < α n < d 2. Pµ n = k = Pµ n = kβ(α n )( + o()) (2.4) Proof. We obtain Theorem 2.7 from (2.4) and from Lemma., if we apply (2.3) both for ξ i and for ξ (r) i.

8 64 I. Fazekas, B. Porvázsnyik We shall use the so called power series distribution. Consider the random variable ξ 0 with the following distribution. Let b 0, b, b 2,... be a sequence of non-negative numbers and let R denote the radius of convergence of the series B(θ) = k=0 b k θ k. k! Assume that R > 0. Let ξ 0 = ξ 0 (θ) have the following distribution p k = p k (θ) = Pξ 0 (θ) = k = Differentiating B(θ) for 0 θ < R, we obtain Eξ 0 (θ) = θb (θ) B(θ), D2 ξ 0 (θ) = θ2 B (θ) B(θ) b kθ k, k = 0,, 2,.... (2.5) k!b(θ) + Eξ 0 (θ) (Eξ 0 (θ)) 2 (see e.g. [7]). We will assume that the distribution of the random variable ξ 0 (θ) satisfies b 0 > 0, b > 0. (2.6) We emphasize that the distribution of ξ 0 = ξ 0 (θ) is not fixed, it depends on θ. We have the following Poisson limit theorem. Theorem 2.8. Suppose that the random variable ξ 0 = ξ 0 (θ) has distribution (2.5), n condition (2.6) is satisfied. Let θ K < R. Let r > and 0. Let r such that p r (θ) λ for some 0 < λ <. Then for all k we have Pµ n = k = λk e λ ( + o()). (2.7) k! Proof. Let k. By the Poisson limit theorem, one has Cp k k r( p r ) k = λk e λ ( + o()). (2.8) k! Relation p r (θ) λ implies that θ = o(), B(θ) = b 0 + o(), B (θ) = b + o() and B (θ) = b 2 + o(). Therefore θ = We know that Eξ 0 = θb (θ) B(θ). Therefore Eξ 0 = b b 0 ( r!(b0 λ + o()) b r ( ) r!(b0λ+o()) /r. b r ) /r ( ) /r ( + o()) = C ( + o()). (2.9) Here and in what follows C denotes an appropriate constant (its value can be different in different formulae). Similarly ( ) /r D 2 ξ 0 = C ( + o()). (2.20)

9 A generalized allocation scheme 65 ow applying condition (2.20), we obtain n r As Eξ (r) 0 = Eξ 0 rp r p r, so (2.9) and condition n r We have We obtain ow applying condition (2.24), we obtain 0, Chebishev s inequality and relations (2.9), P S n = ( + o()). (2.2) 0 imply that ( ) /r Eξ (r) 0 = C ( + o()). (2.22) D 2 ξ (r) 0 = Eξ2 0 a 2 p r ( p r ) 2 + 2arp r ( p r ) 2 r2 p r ( p r ) 2. (2.23) ( ) /r D 2 ξ (r) 0 = C ( + o()). (2.24) n r 0, Chebishev s inequality and relations (2.22), PS (r) k n kr = ( + o()). (2.25) Inserting (2.2), (2.25), and (2.8) into (2.4), we obtain (2.7). 3. Limit theorems for max i η i Let η () = max i η i. η () is the maximal number of balls contained by any of the boxes. Let ξ ( r) 0 be a random variable with distribution Pξ ( r) 0 = k = Pξ 0 = k ξ 0 r. Let ξ ( r) i, i =,...,, be independent copies of ξ ( r) 0. Let S ( r) = ξ( r) i and Eξ ( r) 0 = a r. We can see that a r a. Moreover, a r = a if and only if P(ξ 0 r) =, that is ξ 0 and ξ ( r) 0 have the same distribution. The following representation of η () is useful to obtain limit results. Theorem 3.. We have for all r where P r = Pξ 0 > r. Pη () r = ( P r ) PS( r) n PS n, (3.)

10 66 I. Fazekas, B. Porvázsnyik Proof. Pη () r = Pη r,..., η r = P ξ r,..., ξ r ξ i n P ξ r,..., ξ r, S n = P S n P S n ξ r,..., ξ r = (Pξ r) P S n = ( P r ) PS( r) n PS n. Theorem 3.2. () Let d < a r. Then for all fixed r, as n,, we have Pη () r = ( P r ) ( + o()) (3.2) uniformly for α n < d. (2) Suppose that a r < a and a r < d < d < a. Then for all fixed r, as n,, we have Pη () r = ( P r ) o() (3.3) uniformly for d > α n > d. Proof. () Apply Kolmogorov s law of large numbers for S and S ( r) in (3.). Then (3.2) follows. (2) If d < α n < d and we apply Kolmogorov s law of large numbers, then we obtain lim P S ( r) n,, d <α n <d n = 0. (3.4) Theorem 3.3. Suppose that Eξ0 2 < and let σ r 2 be the variance of ξ( r) 0. Let C <. Then, for all r, as n, such that (α n a) C, we have ( ) Pη () r = ( P r ) Φ C σ r Φ ( ) + o(), for a C r = a, (3.5) σ Pη () r = ( P r ) o(), for a r < a. (3.6) Proof. By the central limit theorem, we have P n = P ξ( r) i a r ξ ( r) i σ r (αn a r ) σ r

11 A generalized allocation scheme 67 ( ) (αn a r ) = Φ + o(). (3.7) σ r In relation (3.) apply (2.) and (3.7) to obtain (3.5) and (3.6). Let η () = max i η i be the maximum in the usual generalized allocation scheme (.). Using large deviation results, we can describe the relation of η () and η (). Introduce notation L r (λ) = Ee λξ( r) 0 where we assume that there exist a positive constant λ ( r) 0 > 0, such that Let Ee λ( r) 0 ξ ( r) 0 < (Cramér s condition). m r (λ) = (ln(l r (λ))), σ 2 r(λ) = m r(λ), 0 λ λ ( r) 0. Let r. If P(ξ 0 = 0) > 0 and P(ξ 0 r) > P(ξ 0 = 0), then ξ ( r) 0 is nondegenerate, therefore m r (.) is a strictly increasing function. Let h = m (α n ), h r = m r (α n ), and β r (α n ) = e h. e h r Theorem 3.4. Assume that r, P(ξ 0 = 0) > 0, and P(ξ 0 r) > P(ξ 0 = 0). Let maxa, a r < d < d 2 < minm(λ 0 ), m r (λ ( r) 0 ). Then, for all r as n,, we have uniformly for d < α n < d 2. Pη () r = Pη () rβ r (α n )( + o()) (3.8) Proof. For the usual generalized allocation scheme, V.F. Kolchin in [7] obtained that Pη () r = ( P r) PS( r) = n (3.9) PS = n for all r where P r = Pξ 0 > r. Using (3.9) and (3.) and applying (2.3) both for ξ i and for ξ ( r) i, the proof of Theorem 3.4 is complete. Theorem 3.5. Suppose that the random variable ξ = ξ(θ) has distribution (2.5), condition (2.6) is satisfied and θ K < R. Let r. Let θ = θ() be such that n p r+ (θ) λ where 0 < λ <. Then, as n, such that 0, we r/(r+) have Pη () = r = e λ + o(), (3.0) Pη () = r + = e λ + o(). (3.)

12 68 I. Fazekas, B. Porvázsnyik Proof. Relation n 0 implies that r/(r+) ( ) /(r+) (r + )!(b0 λ + o()) B(θ) = b 0 + o() and θ =. b r+ Using r + instead of r in the proof of Theorem 2.8, we obtain Let r. Then Moreover, P S n = ( + o()). (3.2) ( ( ) ) /(r+) k r Eξ ( r) k= k b k (r+)!(b0λ+o()) k!b 0 b r+ 0 = ( ( ) ) /(r+) k ( + o()) r b k (r+)!(b0λ+o()) k=0 k!b 0 b r+ ( ) /(r+) = C ( + o()). (3.3) ( ) /(r+) D 2 ξ ( r) 0 C ( + o()). (3.4) Using Chebishev s inequality, (3.3) and (3.4), we obtain Using relations θ 0 and p r+ (θ) λ, we obtain PS ( r) n = ( + o()). (3.5) ( P r ) = o(), ( P r ) = e λ + o(), ( P r+ ) = + o(). (3.6) Inserting (3.2), (3.5), and (3.6) into (3.), we obtain Pη () r = o(), Pη () r = e λ + o(), Pη () r + = + o(). These relations imply (3.0) and (3.). 4. Limit theorems for min i η i In this section we shall prove limit theorems for the minimal content of the boxes. Let η ( ) = min i η i. Let ξ ( r) 0 be a random variable with distribution Pξ ( r) 0 = k = Pξ 0 = k ξ 0 r. Let ξ ( r), i =,...,, be independent copies of ξ ( r) 0. Let S ( r) = ξ( r) i i and Eξ ( r) 0 = a r. One can see that Eξ ( r) 0 Eξ 0 and equality can happen if and only if ξ ( r) 0 = ξ 0. We start with an appropriate representation of η ( )

13 A generalized allocation scheme 69 Theorem 4.. We have for all r where Q r = Pξ 0 < r. Proof. Pη ( ) r = ( Q r ) PS( r) n PS n, (4.) Pη ( ) r = Pη r,..., η r = P ξ r,..., ξ r ξ i n P ξ r,..., ξ r, S n = P S n P S n ξ r,..., ξ r = (Pξ r) P S n = ( Q r ) PS( r) n PS n. Theorem 4.2. Let d < a. Then for all r, as n,, we have uniformly for α n < d. Pη ( ) r = ( Q r ) ( + o()) (4.2) Proof. We apply Kolmogorov s law of large numbers for S Then we obtain (4.2). and S ( r) in (4.). Theorem 4.3. Suppose that Eξ0 2 < and let σ r 2 be the variance of ξ( r) 0. Let C <. Then, for all r, as n, such that (α n a) C, we have ( ) Pη ( ) r = ( Q r ) Φ C σ r Φ ( ) + o(), for a C r = a, (4.3) σ ( ) Pη ( ) r = ( Q r ) Φ ( ) + o(), for a C r > a. (4.4) σ Proof. By the central limit theorem, we have P n = P ξ( r) i a r (αn a r ) ξ ( r) i σ r σ r ( ) (αn a r ) = Φ + o(). (4.5) In relation (4.) apply (2.) and (4.5). Then we obtain (4.3) and (4.4). σ r

14 70 I. Fazekas, B. Porvázsnyik References [] Békéssy, A. On classical occupancy problems. I. Magy. Tud. Akad. Mat. Kutató Int. Közl. 8 (963), o.-2, [2] Chuprunov, A.. and Fazekas, I. Inequalities and strong laws of large numbers for random allocations. Acta Math. Hungar. 09 (2005), o.-2, [3] Chuprunov, A.. and Fazekas, I. An inequality for moments and its applications to the generalized allocation scheme. Publ. Math. Debrecen, 76 (200), no. 3, [4] Chuprunov, A.. and Fazekas, I. One some analogue of the generalized allocation scheme. (Russian) Diskretnaya Matematika, 2(202), o.. [5] Johnson,. L., Kemp, A. W. and Kotz, S. Univariate Discrete Distributions. Third edition. Wiley Series in Probability and Statistics. Wiley-Interscience, Hoboken, J, [6] Kolchin, A. V. and Kolchin, V. F. On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme. (Russian) Diskret. Mat. 8 (2006), no. 4, 3 27; translation in Discrete Math. Appl. 6 (2006), no. 6, [7] Kolchin, V.F. Random Graphs. Cambridge University Press, Cambridge, 999. [8] Kolchin, V.F., Sevast yanov, B.A. and Chistyakov, V.P. Random allocations. V.H. Winston & Sons, Washington D. C [9] Lerner, B., Lone, A. and Rao, M. On generalized Poisson distributions. Probab. Math. Statist. 7 (997), no. 2, Acta Univ. Wratislav. o. 2029, [0] Pavlov, Yu. L. Random Forests. VSP, Utrecht, [] Rozovskii, L. V. Probabilities of large deviations for some classes of distributions satisfying the Cramér condition. (Russian) Zap. auchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 298 (2003), Veroyatn. i Stat. 6, 6 85, 38 39; translation in J. Math. Sci. (. Y.) 28 (2005), no., [2] Rényi, A. Three new proofs and generalization of a theorem of Irving Weiss. Magy. Tud. Akad. Mat. Kutató Int. Közl. 7 (962), o.-2, [3] Weiss, I. Limiting distributions in some occupancy problems. Ann. Math. Statist. 29 (958), o.3,