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

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 ${\lambda }_n=\sum _{i=1}^{n}b(i;n)$, $\lambda =\underset{n\to \infty }{\lim }{\lambda }_n,$ and $m_n=\underset{1\le i\le n}{\max }b(i;n)$. 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.