Poisson Limit Theorems in an Allocation Scheme with an Even Number of Particles in Each Cell

Abstract

We consider an allocation scheme of $2n$ distinguishable particles by $N$ different cells under the condition than each cell contains an even number of particles. We show that this scheme is a general allocation scheme defined by the random variable $\xi_{i}$ with the distribution ${\mathbf{P}}(\xi_{i}=2k)=\frac{\alpha^{2k}}{(2k)!\cosh\alpha},$$k=0,1,2\dots$. Let $\mu_{2r}(N,K,n)$ be a number of cells from the first $K$ cells that contain $2r$ particles. We prove that under some types of convergence of $n,K,N$ to infinity $\mu_{2r}(N,K,n)$ converges in distribution to the Poisson random variable. The limit Poisson random variable is described.