Local limit theorems via Landau-Kolmogorov inequalities

TL;DR

This paper introduces new inequalities between probability metrics and applies them to derive local limit theorems with convergence rates for various models, including the Curie-Weiss model and Erdős-Rényi graphs.

Contribution

It develops novel probability metric inequalities using Landau-Kolmogorov inequalities and applies these to establish new local limit theorems with convergence bounds.

Findings

New inequalities between probability metrics established.

Local limit theorems with explicit convergence rates proved for multiple models.

Enhanced smoothing techniques improve the bounds on convergence.

Abstract

In this article, we prove new inequalities between some common probability metrics. Using these inequalities, we obtain novel local limit theorems for the magnetization in the Curie-Weiss model at high temperature, the number of triangles and isolated vertices in Erd\H{o}s-R\'{e}nyi random graphs, as well as the independence number in a geometric random graph. We also give upper bounds on the rates of convergence for these local limit theorems and also for some other probability metrics. Our proofs are based on the Landau-Kolmogorov inequalities and new smoothing techniques.