This article is cited in
2 papers
On a number of particles in a marked set of cells in a general allocation scheme
A. N. Chuprunov Chuvash State University
Abstract:
In a generalized allocation scheme of
$n$ particles over
$N$ cells we consider the random variable
$\eta_{n,N}(K)$ which is the number of particles in a given set consisting of
$K$ cells. We prove that if
$n, K, N\to\infty$, then under some conditions random variables
$\eta_{n,N}(K)$ are asymptotically normal, and under another conditions
$\eta_{n,N}(K)$ converge in distribution to a Poisson random variable. For the case when
$N\to\infty$ and
$n$ is a fixed number, we find conditions under which
$\eta_{n,N}(K)$ converge in distribution to a binomial random variable with parameters
$n$ and
$s=\frac{K}{N}$,
$0<K<N$, 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
$B(\beta)=\ln(1-\beta)$ the conditions
$n,N,K\to\infty$,
$\frac{K}{N}\to s$,
$N=\gamma\ln(n)+o(\ln(n))$, where
$0< s<1$,
$0<\gamma<\infty$, are satisfied, then distributions of random variables
$\frac{\eta_{n,N}(K)}{n}$ converge to a beta-distribution with parameters
$s\gamma$ and
$(1-s)\gamma$.
Keywords:
generalized allocation scheme, Poisson distribution, Gaussian distribution, binomial distribution, hypergeometric distribution, beta-distribution, local limit theorem.
UDC:
519.212.2+
519.214.5 Received: 27.08.2021
DOI:
10.4213/dm1663
Fulltext:
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