On Mukhin's necessary and sufficient condition for the validity of the local limit theorem

TL;DR

This paper revisits Mukhin's 1984 condition for the local limit theorem, providing a revised proof that establishes a weaker but rigorously proven necessary and sufficient condition with a different formulation.

Contribution

The paper offers a new, rigorous proof of a weaker necessary and sufficient condition for the local limit theorem, differing from Mukhin's original formulation.

Findings

Established a rigorously proven weaker necessary and sufficient condition

Reformulated the condition with a significantly different expression

Provided a concise note revisiting Mukhin's original proof

Abstract

Mukhin found in 1984 an important necessary and sufficient condition for the validity of the local limit theorem. Revisiting the succint proof given in \cite{Mu2}, we could only prove rigorously a weaker necessary and sufficient condition, with a significantly different formulation. This is the object of this short Note.