Explicit constants in the nonuniform local limit theorem for Poisson binomial random variables

Abstract

In a recent paper the authors proved a nonuniform local limit theorem concerning normal approximation of the point probabilities $P(S=k)$ when $S=\sum_{i=1}^{n}X_{i}$ and $X_{1},X_{2},\ldots ,X_{n}$ are independent Bernoulli random variables that may have different success probabilities. However, their main result contained an undetermined constant, somewhat limiting its applicability. In this paper we give a nonuniform bound in the same setting but with explicit constants. Our proof uses Stein’s method and, in particular, the K-function and concentration inequality approaches. We also prove a new uniform local limit theorem for Poisson binomial random variables that is used to help simplify the proof in the nonuniform case.