A generalized allocation scheme

Annales Mathematicae et Informaticae 39 (202) pp. 57 70 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 e-mail: fazekasi@inf.unideb.hu b University of Debrecen, Faculty of Science and Technology Debrecen, Hungary e-mail: 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/-200-0024 project. The project is co-financed by the European Union and the European Social Fund. 57

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 2 + + 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 (.).

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

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)

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)

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

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.

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)

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.)

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

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.)

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 ( )

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

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