Local limit theorems for occupancy models

TL;DR

This paper introduces a general method combining Stein's method for distributional approximation and concentration to prove local limit theorems with good convergence rates for sums of dependent random variables, with applications in germ--grain models and Erdős–Rényi graphs.

Contribution

It develops a versatile approach for establishing local limit theorems with optimal error rates for dependent variables using Stein couplings.

Findings

Proves local CLTs with optimal convergence rates for germ counts.

Establishes local CLTs for degree counts in Erdős–Rényi graphs.

Method applicable to various dependent sum models.

Abstract

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with $d$ neighbours in a germ--grain model, and the number of degree-$d$ vertices in an Erd\H{o}s--R\'enyi random graph. In both cases, the error rate is optimal, up to logarithmic factors.