Error bounds in local limit theorems using Stein's method

TL;DR

This paper develops bounds for local limit theorems using Stein's method, providing new insights into the accuracy of translated Poisson approximations for various discrete distributions.

Contribution

It introduces a general theorem for bounding point probabilities with Stein's method, including a novel bound on the Stein solution for the Poisson distribution.

Findings

Derived optimal convergence rates for the local limit metric

Applied results to Hoeffding CLT, Erdős-Rényi graphs, and Curie-Weiss model

Established new bounds for discrete normal approximation

Abstract

We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in the context of the Hoeffding combinatorial central limit theorem with integer valued summands, of the number of isolated vertices in an Erd\H{o}s-R\'enyi random graph, and of the Curie-Weiss model of magnetism, where we provide optimal or near optimal rates of convergence in the local limit metric. In the Hoeffding example, even the discrete normal approximation bounds seem to be new. The general result follows from Stein's method, and requires a new bound on the Stein solution for the Poisson distribution, which is of general interest.