Extensions of two classical Poisson limit laws to non-stationary independent data

TL;DR

This paper extends classical Poisson limit laws from stationary to non-stationary independent data, broadening their applicability in asymptotic probability theory.

Contribution

It generalizes Poisson limit laws for sums of Bernoulli and geometric random variables to non-stationary independent sequences, beyond the classical i.i.d. setting.

Findings

Extended Poisson limit laws to non-stationary independent data

Provided new asymptotic results for sums of non-stationary Bernoulli and geometric variables

Set the stage for future extensions to dependent data

Abstract

In earlier stages in the introduction to asymptotic methods in probability theory, the weak convergence of sequences $(X_n)_{n\geq 1}$ of Binomial of random variables (\textit{rv}'s) to a Poisson law is classical and easy-to prove. A version of such a result concerning sequences $(Y_n)_{n\geq 1}$ of negative binomial \textit{rv}'s also exists. In both cases, $X_n$ and $Y_n-n$ are by-row sums $S_n[X]$ and $S_n[Y]$ of arrays of Bernoulli \textit{rv}'s and corrected geometric \textit{rv}'s respectively. When considered in the general frame of asymptotic theorems of by-row sums of \textit{rv}'s of arrays, these two simple results in the independent and identically distributed scheme can be generalized to non-stationary data and beyond to non-stationary and dependent data. Further generalizations give interesting results that would not be found by direct methods. In this paper, we focus on generalizations to the non-stationary independent data. Extensions to dependent data will addressed later.