Nonconventional Poisson Limit Theorems

TL;DR

This paper extends classical Poisson limit theorems to more complex sums involving products of Bernoulli variables and applies these results to Markov chains and dynamical systems, revealing new Poissonian limit behaviors.

Contribution

It introduces nonconventional Poisson limit theorems for sums of products of Bernoulli variables and their applications to Markov chains and dynamical systems.

Findings

Extended Poisson limit theorems to sums of products of Bernoulli variables.

Established Poissonian limits for counts of visits in Markov chains.

Applied results to dynamical systems, specifically subshifts of finite type.

Abstract

The classical Poisson theorem says that if $\xi_1,\xi_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\equiv \la/n$ then the sum $S_n=\sum_{i=1}^n\xi_i$ is asymptotically in $n$ Poisson distributed with the parameter $\la$. It turns out that this result can be extended to sums of the form $S_n=\sum_{i=1}^n\xi_{q_1(i)}... \xi_{q_\ell(i)}$ where now $p_n\equiv(\la/n)^{1/\ell}$ and $1\leq q_1(i) <... <q_\ell(i)$ are integer valued increasing functions. We obtain also Poissonian limit for numbers of arrivals to small sets of $\ell$-tuples $X_{q_1(i)},...,X_{q_\ell(i)}$ for some Markov chains $X_n$ and for numbers of arrivals of $T^{q_1(i)}x,...,T^{q_\ell(i)}x$ to small cylinder sets for typical points $x$ of a subshift of finite type $T$.