Classical and Almost Sure Local Limit Theorems

TL;DR

This paper surveys classical and almost sure local limit theorems, discussing their conditions, variants, and recent developments, with a focus on results from Lithuanian and Russian probabilistic research.

Contribution

It provides a comprehensive overview of the conditions, variants, and recent advances in local limit theorems, including almost sure versions, highlighting historical and recent contributions.

Findings

Summary of necessary and sufficient conditions for local limit theorems

Comparison of characteristic function and Bernoulli methods

Coverage of recent almost sure local limit theorems in various stochastic models

Abstract

We present and discuss the many results obtained concerning a famous limit theorem, the local limit theorem, which has many interfaces, with Number Theory notably, and for which, in spite of considerable efforts, the question concerning conditions of validity of the local limit theorem, has up to now no satisfactory solution. These results mostly concern sufficient conditions for the validity of the local limit theorem and its interesting variant forms: strong local limit theorem, strong local limit theorem with convergence in variation. Quite importantly are necessary conditions, and the results obtained are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary condition, and Mukhin's necessary and sufficient condition. Extremely useful and instructive are the counter-examples due to Azlarov and Gamkrelidze, as well as necessary and sufficient conditions obtained for a class of random variables, such as Mitalauskas' characterization of the local limit theorem in the strong form for random variables having stable limit distributions. The method of characteristic functions and the Bernoulli part extraction method, are presented and compared. A second part of the survey is devoted to the more recent study of the almost sure local limit theorem, instilled by Denker and Koch. The almost sure local limit theorems established already cover the i.i.d. case, the stable case, Markov chains, the model of the Dickman function. Our aim in writing this monograph was notably to bring to knowledge many interesting results obtained by the Lithuanian and Russian Schools of Probability during the sixties and after, and which are essentially written in Russian, and moreover often published in Journals of difficult access.