Local limit theorems via Landau–Kolmogorov inequalities

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 Erdős–Rényi 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.