Local Limit Theorems for Poisson's Binomial in the Case of Infinite Expectation

TL;DR

This paper extends local limit theorems for sums of Bernoulli variables to cases with infinite expectation, providing uniform asymptotic results without typical finiteness or smallness assumptions.

Contribution

It generalizes Poisson local limit theorems to infinite expectation scenarios, including dependent variables, with uniform asymptotic results under broad growth conditions.

Findings

Extended local limit theorems to infinite expectation cases.

Results hold uniformly for a wide range of k values.

Applicable to dependent Bernoulli sums, including Sevast'yanov's scheme.

Abstract

Let $ V_{n} = X_{1,n} + X_{2,n} + \cdots + X_{n,n}$ where $X_{i,n}$ are Bernoulli random variables which take the value $1$ with probability $b(i;n)$. Let $\lambda_{n} = \sum\limits_{i=1}^{n} b(i;n) $, $\lambda = \lim\limits_{n \to \infty} \lambda_n,$ and $m_n = \max\limits_{1 \leq i \leq n} b(i;n)$. We derive asymptotic results for $P(V_{n}=k)$ that hold without assuming that $\lambda < +\infty$ or $m_n \to 0$. Also, we do not assume $k$ to be fixed, but instead, our results hold uniformly for all $k$ which satisfy particular growth conditions with respect to $n$. These results extend known Poisson local limit theorems to the case when $\lambda = +\infty$. While our results apply to triangular arrays, without the assumption that \(m_n \to 0\) they continue to hold for sums of Bernoulli random variables. In this setting, our growth conditions cover a range of values for $k$ not centered at $\lambda_n$, thus complementing known local limit theorems based on approximation by the normal distribution. In addition, we show that our local limit theorems apply to a scheme of dependent random variables introduced in the work of Sevast'yanov.