Inequalities and limit theorems for random allocations

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1 doi: /v A A L E S U I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I P O L O I A VOL. LXV, O., 20 SECTIO A ISTVÁ FAZEKAS 2, ALEXEY CHUPRUOV and JÓZSEF TÚRI Inequalities and limit theorems for random allocations Abstract. Random allocations of balls into boxes are considered. Properties of the number of boxes containing a fixed number of balls are studied. A moment inequality is obtained. A merge theorem with Poissonian accompanying laws is proved. It implies an almost sure limit theorem with a mixture of Poissonian laws as limiting distribution. Almost sure versions of the central limit theorem are obtained when the parameters are in the central domain.. Introduction. Let n balls be placed successively and independently into boxes. Let µ r n, denote the number of boxes containing r balls. There are several theorems concerning the limit laws of µ r n, when the parameters belong to certain domains see e.g. Weiss [6], Rényi [4], Békéssy [2], and the monograph Kolchin Sevast yanov Chistyakov [2]. It is known that if n, in the central domain, then the limit of the standardized µ r n, is standard normal. In the left-hand and in the right-hand r-domains the limit of µ r n, is Poisson distribution. Strong laws of large numbers are obtained for µ r n, in Chuprunov Fazekas [5]. Concerning the generalized allocation scheme see Kolchin []. Corresponding author. 2 Supported by the Hungarian Foundation of Scientific Researches under Grant o. OTKA T Mathematics Subject Classification. 60F05, 60F5, 60C05. Key words and phrases. Random allocation, moment inequality, merge theorem, almost sure limit theorem.

2 70 I. Fazekas, A. Chuprunov, J. Túri In this paper the most general result is the inequality in Theorem 2.. It gives an upper bound for the L 2 -distance of µ r n, and its conditional expectation given the last n k allocations. Then asymptotic results are considered. The most interesting case is the Poisson-type limiting distribution. In that case we do not have one single limiting distribution. Instead of a limit theorem we can prove a merge theorem, i.e. we can give a family of accompanying Poissonian laws being close to the original distributions Theorem 2.2. Then we obtain almost sure a.s. versions of the limit theorems for µ r n,. The general form of the a.s. limit theorem is the following. Let Y n, n be a sequence of random elements defined on the probability space Ω, A, P. A.s. limit theorems state that n. d k δ Yk ω ν, D n k= as n, for almost every ω Ω, where δ x is the unit mass at point x and ν denotes weak convergence to the probability measure ν. In the simplest form of the a.s. CLT Y k = X + + X k / k, where X, X 2,..., are i.i.d. real random variables with mean 0 and variance, d k = /k, D n = log n, and ν is the standard normal law 0, ; see Berkes [3] for an overview. Recently, several papers are devoted to the background, the general forms and certain special cases of the a.s. limit theorem, see e.g. Berkes Csáki [4], Fazekas and Rychlik [8], Matuła [3], Hörmann [0], Orzóg Rychlik [5]. The present paper can be considered as an extension of some results in the paper of Fazekas Chuprunov [6], where a.s. limit theorems were obtained for the number of empty boxes see also Becker Kern []. In Section 2, we consider an appropriate representation of µ r n, in terms of independent, uniformly distributed random variables in order to handle the dependence structure inside the array µ r n,, n, =, 2,.... As µ r n, depends on two parameters, we consider a.s. limit theorems of the form.2 d kk δ D YkK ω ν, n k,k T n as n,, for almost every ω Ω, where T n denotes a two-dimensional domain. To prove the above type theorems, we apply a general a.s. limit theorem, i.e. Theorem 2. of Fazekas Chuprunov [6]. We quote it in Theorem 4.. This result is an extension of known general a.s. limit theorems see e.g. Fazekas and Rychlik [8]. We remark that multiindex versions of a.s. limit theorems were obtained in Fazekas Rychlik [9]. However, as the weights there are of product type, we can not apply those results for domains like {k, i : α k i α 2 k, k }. In this paper we use the general theorem to obtain Theorems 2.3, 2.4, 2.5, and 2.6. Among them Theorems 2.5, 2.6 concern the central domain

3 Inequalities and limit theorems for random allocations 7 i.e. when 0 < α n/ α 2 < and the limiting distribution is standard normal. In Theorem 2.4 the parameters can vary in a domain not included in the central domain but the limiting distribution is again standard normal. The most interesting case is the Poisson-type limiting distribution Theorem 2.3. The limiting distribution in the almost sure limit theorem i.e. in Theorem 2.3 will be a mixture of the accompanying laws in the usual limit theorem i.e. in Theorem 2.2. In almost sure limit theory the above situation is well-known see Fazekas Chuprunov [7] for semistable laws, see also Theorems 2.0, 2,, 2.2 in Fazekas Chuprunov [6] for random allocations. 2. Main results. Random allocations. Let ξ, ξ j, j, be independent random variables uniformly distributed on [0, ]. Let. Consider the subdivision of the interval [0, into the subintervals i = i = [ i, i, i. We consider the intervals i, i =,...,, as a row of boxes. Random variables ξ j, j =, 2,..., are realizations of ξ. Each realization of ξ we treat as a random allocation of one ball into the boxes. The event ξ j i means that the jth ball falls into the ith box. Let n, A 0 = {, 2,..., n}. 2. µ r n, = I {ξj i } I {ξj / i } i= A =r, j A j A 0 \A A A 0 is the number of boxes containing r balls and Cn r n r r is its expectation. Here Cn r = n r is the binomial coefficient and IB is the indicator of the event B. For n, we will use the notation α = n and p rα = α r /r!e α. It is known see Kolchin et al. [2], Ch. 2, Sec., Theorem that the following limit relations 2.2 and 2.3 hold for any fixed r, t and if n, such that α = o. For the expectation we have 2.2 Eµ r n, = p r α + p r α r α/2 C 2 r /α + O / and for the covariances we have 2.3 covµ r n,, µ t n, σ rt α, where σ rr α = p r α p r α p r αα r 2 /α, σ rt α = p r αp t α + α rα t/α, if t r. We shall use the notation D r n, = D 2 µ r n, = covµ r n,, µ r n,.

4 72 I. Fazekas, A. Chuprunov, J. Túri We shall need a lower bound for D r n,, therefore the following remark will be useful. Remark 2.. α r2 p r α p r α α c r > 0, if r 2 is fixed and α is arbitrary, or if r = 0, and α α 0 > 0. As in the theory of random allocations the roles of n and are fixed, therefore we shall use the following notation for two-dimensional indices: n,, k, K 2. Let S r n = µ rn, Eµ r n, D r n, be the standardized variable, where n, 2. The main inequality. Let n,, r, 0 k n. Recall that n is the number of balls, is the number of boxes. ξ j denotes the jth ball, i denotes the ith box. We use the notation A k = {k +,..., n}, k = 0,,..., n. Let ζ n = ζ n = I {ξj i } i= A =r, j A j A 0 \A A A 0 I {ξj / i } C r n r n r. We see that ζ n = µ r n, Eµ r n,, c.f. 2.. We have ζ n = i= A =r, A A 0 η ia Eη ia, where η ia = I {ξj i } j A j A 0 \A I {ξj / i } is the indicator of the event that the ith box contains the balls with indices in the set A and it does not contain any other ball. Let F kn be the σ- algebra generated by ξ k+,..., ξ n. We will use the following conditional

5 Inequalities and limit theorems for random allocations 73 expectations η k ia = Eη ia F nk and ζn k = ζn k = Eζ n F kn = 2.4 = i= A =r, A A 0 i= r A A k A =r, A A 0 I {ξj i } j A A k j A k \A η k ia Eηk ia k r A Ak I {ξj / i } r n r. The following inequality will play an important role in the proofs of our theorems. Theorem 2.. Let 0 < k < n, 0 < r n and be fixed. Then we have [ 2.5 Eζ n ζn k 2 ckα r n+k α r + ] n r α +, where c < does not depend on n,, and k, but can depend on r. Remark 2.2. In Fazekas Chuprunov [6] the following inequality was obtained for the number of empty boxes. Let r = 0. Let k < n and be fixed. Then we have 2.6 Eζ n ζn k 2 k n k and 2.7 Eζ n ζ k n 2 kn. In Chuprunov Fazekas [6] a fourth moment inequality was obtained for µ r n,. Limit theorems for random allocations for r 2. First we consider the Poisson limiting distribution. In that case we do not have one single limiting distribution in the ordinary limit theorem. Instead of a limit theorem we can prove a merge theorem, i.e. we can give a family of accompanying laws being close to the original distributions Theorem 2.2. The limiting distribution in the almost sure limit theorem i.e. in Theorem 2.3 will be a mixture of the accompanying laws. The following result is a version of Theorem 3 in Section 3, Chapter II of Kolchin Sevast yanov Chistyakov [2]. In our theorem the novelty is that we state uniformity with respect to n, in a certain domain, while l remains fixed.

6 74 I. Fazekas, A. Chuprunov, J. Túri Theorem 2.2. Let r 2 and l be fixed. Then, as n,, 2.8 Pµ r n, = l = l! p r l e pr + o uniformly with respect to the domain T = {n, : n 2r /2r 2 log n}. ow turn to the a.s. version of Theorem 2.2. Theorem 2.3. Let r 2, 0 < λ < λ 2 < be fixed. Let T n be the following domain in 2 { T n = k, K 2 : k n, λ k } λ 2. Let Q n ω = Then, as n, K r r r λ 2 λ log n k,k T n K 2 r Q n ω µ τ δ µrk,kω. for almost all ω Ω, where τ is a random variable with distribution λ2 x r l 2.9 Pτ = l = e xr r! dx, l = 0,,.... λ 2 λ λ l! r! ow consider the case of the normal limiting distribution. Theorem B. Let r 2 be fixed. If n,, so that p r α, then S r n γ. Here and in the following γ denotes the standard normal law. The proof of Theorem B can be found in the monograph Kolchin et al. [2], Ch. 2, Sec. 3, Theorem 4. Consider an almost sure version of Theorem B. Theorem 2.4. Let r 2 be fixed, 0 < α, α 2 < and { T n = k, K 2 : k n, α k K α 2 k 2r+/2r}. Let Q r+ n ω = log n Then, as n, we have klog α 2 log α + /2r log kk δ S r kk ω. k,k T n Q r+ n ω γ, for almost every ω Ω.

7 Inequalities and limit theorems for random allocations 75 Almost sure limit theorems for random allocations in the central domain. If n,, so that 0 < α n α 2 <, where α and α 2 are some constants, then it is said that n, in a central domain. In a central domain we have the following central limit theorem. Theorem A. Let 0 < α < α 2 <. If n,, so that α = n [α, α 2 ], then S r n γ. The proof of Theorem A can be found in the monograph Kolchin et al. [2], Ch. 2, Sec. 2, Theorem 4. Consider almost sure versions of Theorem A. In the following theorems the domain is narrower than the one in Theorem 2.4, but they are valid for arbitrary r 0. Theorem 2.5. Let r 0 be fixed, 0 < α < α 2 < and Q r n ω = Then, as n, we have log α 2 log α log n k n {K : α k K α 2} Q r n ω γ, for almost every ω Ω. kk δ S r kk ω. In the above theorem the limit was considered for n and the indices of the summands were in a fixed central domain. The following theorem is a two-index limit theorem, i.e. n and. The relation of n and could be arbitrary, however, as the indices of the summands are in a fixed central domain, we assume that n, is in the central domain considered. Theorem 2.6. Let r 0 be fixed, 0 < α < α 2 < and Q r n ω = log α 2 log α log n k n {K : K, α k K α 2} Then, as n,, so that α n α 2, we have Q r n ω γ, for almost every ω Ω. kk δ S r kk ω. 3. Proof of Theorem 2.. Since Eη ia = Eη k ia and Eη i A η k i A η i2 A 2 η k i 2 A 2 = Eη i A η i2 A 2 Eη k i A η k i 2 A 2,

8 76 I. Fazekas, A. Chuprunov, J. Túri for any A, A 2, we have Eζ n ζn k 2 = = i,i 2 = i= A =r, A A 0 η ia η k ia k Eη ia ηi A η i2 A 2 η k A = A 2 =r, A,A 2 A 0 = i i i i 2 A A 2, A = A 2 =r, A,A 2 A 0 i= i= A A 2 =, A = A 2 =r, A,A 2 A 0 A A 2, A = A 2 =r, A,A 2 A 0 A =r, A A 0 = B + B 2 + B 3 + B 4. 2 i 2 A 2 Eη i A η i2 A 2 Eη k i A η k i 2 A 2 Eη i A η i2 A 2 Eη k i A η k i 2 A 2 Eη ia η ia2 Eη k ia η k ia 2 Eη ia 2 Eη k ia 2 First consider B. Let i i 2, A A 2 and j A A 2. Then I {ξj i }I {ξj i2 } = 0, therefore Eη i A η i2 A 2 = 0. So B 0. ow turn to B 3. ow i = i 2, A A 2. If j A \ A 2 or j A 2 \ A, then I {ξj i }I {ξj / i2 } = 0 or I {ξj / i }I {ξj i2 } = 0. So Eη i A η i2 A 2 = 0. Therefore we have B 3 0.

9 Inequalities and limit theorems for random allocations 77 ow consider B 2. Let i i 2 and A A 2 =. It holds that Eη i A η i2 A 2 Eη k = 2r 2 2r = 2r i A η k i 2 A 2 n 2r 2k 2r Ak A A k A 2 2 n k Ak A A k A 2 2 n 2r 2k 2r+x 2 n k x. Here x = A k A + A k A 2, so we have 0 x 2r, n k. ow let a = 2, b =. Then 0 < a < b <, moreover b 2 a = / 2. First consider those terms from B 2 in which x = 2r. It means that A, A 2 A k. The number of these terms is n k!/r!r!n k 2r!. The magnitude of these terms is 2 n 2r 2k 2 n k 2r = 2r an k 2r a k b 2k 2r an k 2r kb 2k 2. 2r Above we applied the mean value theorem. So the contribution of these terms is not greater than n k! r!r!n k 2r! 2r 2 n k 2r k 2k 2 = B 2. ow turn to the remaining terms of B 2, i.e. the terms with x < 2r. The number of these terms is n! r!r!n 2r! n k! 2rkn2r r!r!n k 2r! r!r! = B 22. Above we applied the following fact. If 0 b i a i c and a i b i l for i =, 2,..., s, then s i= a i s i= b i slc s. Using the mean value

10 78 I. Fazekas, A. Chuprunov, J. Túri theorem, we obtain for the magnitudes of these terms that a n 2r 2r b 2k 2r+x a n k x = an k 2r 2r an k 2r 2r = 2r = B Therefore we have a k b 2k + b 2k a b 2r x 2k k 2 + b2k 2r x n k 2r 2k k 2 + 2k 2r x 3. B 2 B 2 + B 22 B 222 c n2r 2r n+k 2r 2 k + c n2r 2r + c n2r 2r n+k 2r 2 k 2 n+k 2r k cα 2r n+k kα +. Finally, consider B 4. Let r = {, 2,..., k} A = r A A k. We have B 4 = A =r, A A 0 = = A =r, A A 0 Eη ia 2 Eη k ia 2 r min{k,r} r =max{r n k,0} n r 2r 2k r C r k Cr r n k r r r n r n k r r r+r n+k r r

11 = Inequalities and limit theorems for random allocations 79 min{k,r} C r k Cr r n k r r =max{r n k,0} min{k,r} r =max{r n k,0} r r =0 k r r! n r r k r r r! n r r r! r r! r n r r k r r n r r k r n r r k min{r,k}. Separating the term with r = 0, then applying the mean value theorem, we obtain r k r n r r B 4 r r =! r r! r n r + nr r! r n r k 3.2 n r r kα r kα r r = n r e + α r! k r n r! + αr r!. ow, inequalities 3. and 3.2 imply Proofs of the limit theorems. k n r Proof of Theorem 2.2. Consider i.i.d. random variables η, η 2,..., η having Poisson distribution with parameter α. Let ζ = η + + η. Consider also i.i.d. random variables η r, ηr 2,..., ηr having the following distribution = l = Pη i = l η i r. Let ζ r = ηr Pη r i 4. Pµ r n, = l = + + η r. By Lemma at page 60 of Kolchin et al. [2] p lr Pζr l l p r = n lr = F G l Pζ = n H, say. On the domain T, as n,, we have α 0 and p r α 0. Therefore, concerning F, we have l p l r p r l pr l! p. r l e pr e pr Taking logarithm, then applying Taylor s expansion, we obtain p r e pr as n, uniformly in T.

12 80 I. Fazekas, A. Chuprunov, J. Túri To handle G, we need the following result Theorem on p. 6 of Kolchin et al. [2]. For r 2, as m, so that αm, we have Pζ r m = t = uniformly with respect to t mαr σ r m α r = Eη r i Therefore = α rp r p r, σ 2 r = D 2 η r i = t mαr 2 e 2mσr 2 + o σ r 2πm in any finite interval. Here α p r 2 p r α p r 2 p r. α G = Pζ r n lr lαr l = n lr = 2 e 2 lσr 2 + o. σ r 2π l By straightforward calculations we obtain G / 2π lα / 2πn uniformly in T. Finally, turn to H. As ζ has Poisson distribution, applying the Stirling formula, we obtain H = Pζ = n = nn n! e n 2πn uniformly. Substituting the asymptotic values of F, G, H into 4., we obtain 2.8. The proofs of our a.s. limit theorems are based on the following general a.s. limit theorem for two-dimensional domains see Theorem 2. of Fazekas Chuprunov [6]. Actually the theorem is a version of Theorem. in Fazekas Rychlik [8]. Let {α k} and {α 2 k} be given integer valued sequences with α k α 2 k <, for k. Let B, ϱ be a complete separable metric space and let ζ ki, α k i α 2 k, k be an array of random elements in B. Let µ ζ denote the distribution of the random element ζ. Let log + x = log x, if x and log + x = 0, if x <. Theorem 4.. Assume that there exist C > 0, ε > 0; an increasing sequence of positive numbers c n with lim n c n =, c n+ /c n = O; and B-valued random elements ζlj ki, for k, i, l, j, k < l, α k i α 2 k, α l j α 2 l such that the random elements ζ ki and ζlj ki are independent for k < l and for any i, j; and 4.2 E{ϱζ lj, ζ ki lj } C c k/c l β, for k < l and for any i, j, where β > 0. Let 0 d k logc k+ /c k, assume that k= d k =. Assume that d k = α 2 k i=α k d ki

13 Inequalities and limit theorems for random allocations 8 for each k, with nonnegative numbers d ki. Let D n = n k= d k. Then for any probability distribution µ on the Borel σ-algebra of B the following two statements are equivalent 4.3 n α 2 k D n k= i=α k for almost every ω Ω; d ki δ ζki ω µ, as n, 4.4 n α 2 k D n k= i=α k d ki µ ζki µ, as n. Remark 4.. If condition 4.2 is valid only for < k 0 k < l, then Theorem 4. remains valid. ow we can turn to the proofs of the a.s. limit theorems. Proof of Theorem 2.3. Let ζ kk = µ r k, K. For k < n let ζn kk = ζk n + Eζn, k where ζn k is defined in 2.4. We show that ζn kk satisfies the conditions of Theorem 4.. ζn kk and ζ kk are independent for k < n. By Theorem 2., we have 2 n r E ζ n ζn kk k n r k c0 k c0 c0 n /r n λ 2 r because n, T n. Therefore d k = c k is an appropriate choice for any k positive constant c. Let d kk = for k, K with λ K 2 /r λ K /r 2. Then d k = d kk = K 2 /r r r λ 2 λ k. k= {K : k/λ 2 r r K k/λ r r } Therefore the above choice is possible. So, in Theorem 4., we can put n n r D n = d k = r λ 2 λ k r r λ 2 λ log n. k= ow we remark that we can apply Theorem 2.2 because the domain in that theorem is wider that the one in Theorem 2.3. According to Theorem 4., we have to prove that 4.5 F = r rλ 2 λ log n Pτ = l n { } k= K 2 K : λ k K /r λ r 2 Pµ r k, K = l

14 82 I. Fazekas, A. Chuprunov, J. Túri where τ is defined in 2.9. It is easier to calculate F in a wider domain and then remove the surplus, that is n F = { K= k : λ k K /r λ 2 } n/λ r r K=n/λ 2 r r say, where n = n/λ r r. ow consider the following approximations. k, K, so that λ k K /r Kp r = K αr r! e α = K r! Therefore we obtain l! {k : λ k K /r λ 2} l! l! So we have A K r {k : λ k K /r λ K r 2} λ2 λ x r r! K r r λ 2 k=n = A B, Since α = k/k 0 if λ 2, therefore e α e 0 =. So we have k r e k/k k r K r! K /r. Kp r l e Kpr l e xr r! dx. r r λ 2 λ ln n λ 2 λ l! For B we have λ2 λ n K= x r r! 0 B c log n K l! k r l exp k r r! K r r! K r l e xr r! dx. {k : λ k K /r λ 2} n/λ r r K=n/λ 2 r r K r r λ 2 k=n K r K 2 r 0 Kp r l e Kpr as n. So the limit of F is the same as the limit of A. It proves 4.5. Proof of Theorem 2.4. Let r 2 be fixed. Let ζ kk = S r kk. For k < n let ζn kk = ζk n/d r n, where ζk n is defined in 2.4. We show that ζn kk satisfies the conditions of Theorem 4.. ζn kk and ζ kk are independent for k < n.

15 Inequalities and limit theorems for random allocations 83 As r 2, by 2.3 and Remark 2., Cα r e α D r n 2, where C > 0. Therefore, by Theorem 2., we have E ζ n ζ kk n 2 c0 k n αr+ + c k n, if n, T n,. Therefore d k = c k is an appropriate choice for any positive constant c. ow let d k,k = k log α 2 log α + 2r log k K. Then we have {K : α k K α 2 k +2r/2r } d k,k k = d k. So, in Theorem 4., we can put D n = log n. If n,, so that n, T n,, then p r α. So we can apply Theorem B. We obtain log n k,k T n, d kk µ S r kk as n. So we can apply Theorem 4.. γ, Proof of Theorem 2.5. For r = 0 our result is Theorem 2.4 of Fazekas and Chuprunov [6]. ow let r. Let ζ kk = S r kk. For k < n let ζn kk = ζk n/d r n, where ζk n is defined in 2.4. We will show that ζn kk satisfies the conditions of Theorem 4.. ζn kk and ζ kk are independent for k < n. By 2.3 and Remark 2., in the central domain C D r n 2, where C depends only on α and α 2. Therefore, by Theorem 2., we have 2 E ζ n ζn kk k c0 D r c 0 k n 2 C c 0α 2 k C n. Therefore d k = c k Moreover, as d k = k is an appropriate choice for any positive constant c. {K : k α 2 K k α } K k log α 2 log α, the above choice is possible. So, in Theorem 4., we can put D n = log α 2 log α log n. By Theorem A, log α 2 log α log n kk µ S r γ, kk k n {K : α k K α 2} as n. So we can apply Theorem 4..

16 84 I. Fazekas, A. Chuprunov, J. Túri Proof of Theorem 2.6. Consider Q r n difference is Q r n ω Q r n ω= log α 2 log α log n from Theorem 2.5 and Q r n. Their k n{k : K>, α k K α 2} kk δ S r ω. kk As the summands are probability measures, we can confine attention to the weights. However, a direct calculation shows that kk clog α 2 log α 2. k n {K : K>, α k K α 2} Therefore, when for a fixed ω we have Q r n ω γ, as n, then Q r n ω γ, as n,. References [] Becker-Kern, P., An almost sure limit theorem for mixtures of domains in random allocation, Studia Sci. Math. Hungar. 44, no , [2] Békéssy, A., On classical occupancy problems. I, Magy. Tud. Akad. Mat. Kutató Int. Közl , [3] Berkes, I., Results and problems related to the pointwise central limit theorem, Szyszkowicz, B., Ed., Asymptotic Results in Probability and Statistics, Elsevier, Amsterdam, 998, [4] Berkes, I., Csáki, E., A universal result in almost sure central limit theory, Stoch. Proc. Appl , [5] Chuprunov, A., Fazekas, I., Inequalities and strong laws of large numbers for random allocations, Acta Math. Hungar. 09, no , [6] Fazekas, I., Chuprunov, A., Almost sure limit theorems for random allocations, Studia Sci. Math. Hungar. 42, no , [7] Fazekas, I., Chuprunov, A., An almost sure functional limit theorem for the domain of geometric partial attraction of semistable laws, J. Theoret. Probab. 20, no , [8] Fazekas, I., Rychlik, Z., Almost sure functional limit theorems, Ann. Univ. Mariae Curie-Skłodowska Sect. A , 8. [9] Fazekas, I., Rychlik, Z., Almost sure central limit theorems for random fields, Math. achr , 2 8. [0] Hörmann, S., An extension of almost sure central limit theory, Statist. Probab. Lett. 76, no , [] Kolchin, A. V., Limit theorems for a generalized allocation scheme, Diskret. Mat. 5, no , Russian; English translation in Discrete Math. Appl. 3, no , [2] Kolchin, V. F., Sevast yanov, B. A. and Chistyakov, V. P., Random Allocations, V. H. Winston & Sons, Washington D. C., 978. [3] Matuła, P., On almost sure limit theorems for positively dependent random variables, Statist. Probab. Lett. 74, no. 2005, [4] Rényi, A., Three new proofs and generalization of a theorem of Irving Weiss, Magy. Tud. Akad. Mat. Kutató Int. Közl , [5] Orzóg, M., Rychlik, Z., On the random functional central limit theorems with almost sure convergence, Probab. Math. Statist. 27, no. 2007,

17 Inequalities and limit theorems for random allocations 85 [6] Weiss, I., Limiting distributions in some occupancy problems, Ann. Math. Statist , István Fazekas Faculty of Informatics University of Debrecen P.O. Box 2, 400 Debrecen Hungary Alexey Chuprunov Department of Math. Stat. and Probability Kazan State University Universitetskaya 7, Kazan Russia József Túri Faculty of Mechanical Engineering and Informatics University of Miskolc 355 Miskolc-Egyetemváros Hungary Received March, 200