A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

TL;DR

This paper introduces a novel probabilistic method to derive sharp bounds on the total variation distance for sums of locally dependent random variables, improving approximation error estimates for Poisson and compound Poisson distributions.

Contribution

The paper presents a new probabilistic approach that yields sharper bounds on approximation errors, incorporating a smoothness factor of order O(σ^{-2}), for sums of dependent variables.

Findings

Provides bounds with a smoothness factor depending on variance

Achieves sharper error estimates for a wide parameter range

Includes examples with rare events in Bernoulli sequences

Abstract

Let $X_1,X_2,...,X_n$ be a sequence of independent or locally dependent random variables taking values in $\mathbb{Z}_+$. In this paper, we derive sharp bounds, via a new probabilistic method, for the total variation distance between the distribution of the sum $\sum_{i=1}^nX_i$ and an appropriate Poisson or compound Poisson distribution. These bounds include a factor which depends on the smoothness of the approximating Poisson or compound Poisson distribution. This "smoothness factor" is of order $\mathrm{O}(\sigma ^{-2})$, according to a heuristic argument, where $\sigma ^2$ denotes the variance of the approximating distribution. In this way, we offer sharp error estimates for a large range of values of the parameters. Finally, specific examples concerning appearances of rare runs in sequences of Bernoulli trials are presented by way of illustration.