Local limit theorems and mod-phi convergence

TL;DR

This paper establishes local limit theorems for sequences of random variables converging mod-phi, providing new proofs and identifying scales of stable approximation, with applications across various mathematical fields.

Contribution

It introduces two novel proofs of local limit theorems for mod-phi convergent sequences, enhancing understanding of stable approximation scales.

Findings

New proofs based on zone of control and L1 convergence methods

Identification of infinitesimal scales for stable approximation

Application to diverse fields like number theory and statistical mechanics

Abstract

We prove local limit theorems for mod-{\phi} convergent sequences of random variables, {\phi} being a stable distribution. In particular, we give two new proofs of a local limit theorem in the framework of mod-phi convergence: one proof based on the notion of zone of control, and one proof based on the notion of mod-{\phi} convergence in L1(iR). These new approaches allow us to identify the infinitesimal scales at which the stable approximation is valid. We complete our analysis with a large variety of examples to which our results apply, and which stem from random matrix theory, number theory, combinatorics or statistical mechanics.