On a number of particles in a marked set of cells in a general allocation scheme

In a generalized allocation scheme of n particles over N cells we consider the random variable η n , N ( K ) which is the number of particles in a given set consisting of K cells. We prove that if n , K , N → ∞, then under some conditions random variables η n , N ( K ) are asymptotically normal, and under another conditions η n , N ( K ) converge in distribution to a Poisson random variable. For the case when N → ∞ and n is a fixed number, we find conditions under which η n , N ( K ) converge in distribution to a binomial random variable with parameters n and s = <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfrac> <m:mi>K</m:mi> <m:mi>N</m:mi> </m:mfrac> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array K/N array , 0 < K < N , multiplied by a integer coefficient. It is shown that if for a generalized allocation scheme of n particles over N cells with random variables having a power series distribution defined by the function B ( β ) = ln(1 − β ) the conditions n , N , K → ∞, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfrac> <m:mi>K</m:mi> <m:mi>N</m:mi> </m:mfrac> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array K/N array → s , N = γ ln( n ) + o (ln( n )), where 0 < s < 1, 0 < γ < ∞, are satisfied, then distributions of random variables <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mtable rowspacing="4pt" columnspacing="1em"> <m:mtr> <m:mtd> <m:mstyle displaystyle="true"> <m:mfrac> <m:mrow> <m:msub> <m:mi>η</m:mi> <m:mrow class="MJX-TeXAtom-ORD"> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> </m:mrow> </m:msub> <m:mo stretchy="false">(</m:mo> <m:mi>K</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:mfrac> </m:mstyle> </m:mtd> </m:mtr> </m:mtable> </m:math> array ηn,N(K)n array converge to a beta-distribution with parameters sγ and (1 − s ) γ .