A generalized allocation scheme
István Fazekas, Bettina Porvázsnyik · University of Debrecen Electronic Archive (University of Debrecen) · 2012
The generalized allocation scheme was introduced by V.F.Kolchin [1].Let ξ1, ξ2, . . ., ξN be independent identically distributed non-negative integer valued non-degenerate random variables.Consider the random variables η 1 , . . ., η N with joint distributionLet ξi have Poisson distribution, then (η 1 , . . ., η N ) has polynomial distribution.Therefore {η 1 = k1, . . ., η N = kN } means that the contents of the boxes are k1, . . ., kN after allocating n balls into N boxes during the usual allocation procedure.Our aim is to study random variables η1, . . ., ηN with joint distributionIt can be considered as a general allocation scheme when we place at least n balls into N boxes.Let µnN denote the number of cases when {ηi = r}.That is µnN is the number of boxes containing r balls.We shall prove limit theorems for P{µnN = k}.Moreover, we shall consider the asymptotic behaviour of P{max 1≤i≤N ηi ≤ r} and P{min 1≤i≤N ηi ≤ r}.