Abstract:
We consider a random variable $\mu_r(n, K, N)$ being the number of cells containing $r$ particles among first $K$ cells in an equiprobable allocation scheme of at most $n$ distinguishable particles over $N$ different cells. We find conditions ensuring the convergence of these random variables to a random Poisson variable. We describe a limit distribution. These conditions are of a simplest form, when the number of particles $r$ belongs to a bounded set or as $K$ is equivalent to $\sqrt{N}$. Then random variables $\mu_r(n, K, N)$ behave as the sums of independent identically distributed indicators, namely, as binomial random variables, and our conditions coincide with the conditions of a classical Poisson limit theorem. We obtain analogues of these theorems for an equiprobable allocation scheme of $n$ distinguishable particles of $N$ different cells. The proofs of these theorems are based on the Poisson limit theorem for the sums of exchangeable indicators and on an analogue of the local limit Gnedenko theorem.
Keywords:allocation scheme of distinguishable particles over different cells, Poisson random variable, Gaussian random variable, limit theorem, local limit theorem.
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