Invariance principle for numbers of particles in cells of a general allocation scheme

I. Fazekas ; A. N. Chuprunov

Diskretnaya Matematika, Tome 35 (2023) no. 3, pp. 81-99

Résumé

Let $\eta_1,\dots\eta_N$ be a generalized allocation scheme of $n$ particles over $N$ cells defined by independent random variables $\xi_1,\dots,\xi_N$ having power series distribution with parameter $\beta$. Denote by $m(\beta)$ and $\sigma^2(\beta)$ an expectation and a variance of $\xi_i$. Let $\beta$ be such that $\frac{n}{N}=m(\beta)$. We consider random processes $X_{n,N}(t)=\sum_{i=1}^{[tN]}\eta_i$ and $Y_{n,N}(t)=n^{-1/2}(X_{n,N}(t)-[tN]\frac{n}{N})$, $0\le t\le 1$. We find conditions under which for $n,N\to\infty$ the random processes $\sigma_{-1}(\beta)\sqrt{\frac{n}{N}}Y_{n,N}$ converge in distribution in the Skhorohod space to a Brounian bridge, and conditions ubder which for fixes $n$ and $N\to\infty$ the random processes $X_{n,N}$ converge in distribution in the Skhorohod space to $nF_n$, where $F_n$ is an empirical process.

Détail

Keywords: invariance principle, generalized allocation scheme, Poisson limit theorem, local limit theorem, empirical process, Brownian bridge.