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 Ufimsk. Mat. Zh., 2020, Volume 12, Issue 3, Pages 3–10 (Mi ufa520)

Poisson limit theorems in allocation schemes of distinguishable particles

F. A. Abdushukurov

Institute of Computational Mathematics and Information Technologies, Kazan Federal University, Kremlevskaya str., 35, 420008, Kazan, Russia

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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English version:
Ufa Mathematical Journal, 2020, 12:3, 3–10 (PDF, 357 kB); https://doi.org/10.13108/2020-12-3-3

Bibliographic databases:

UDC: 517.958
MSC: 60C05, 60F05

Citation: F. A. Abdushukurov, “Poisson limit theorems in allocation schemes of distinguishable particles”, Ufimsk. Mat. Zh., 12:3 (2020), 3–10; Ufa Math. J., 12:3 (2020), 3–10

Citation in format AMSBIB
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