## Classical and almost sure local limit theorems

### Volume 589 / 2023

#### Abstract

In this monograph, we present and discuss numerous results obtained concerning a famous limit theorem, the local limit theorem (LLT), 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 LLT and its interesting variant forms: strong LLT and strong LLT with convergence in variation. Quite important are necessary conditions, and the results obtained are sparse: Rozanov’s necessary condition, Gamkrelidze’s necessary condition, and, almost isolated among the flow of results, 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 LLT 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. The study of the LLT (old and recent results) consists of three parts:

The LLT for sums of independent, identically distributed random variables: Gnedenko’s theorem, Ibragimov and Linnik’s characterization of the speed of convergence under moments conditions, stronger forms, Galstyan’s results, strong LLTs with convergence in variation, versions for densities, the case of weighted sums of i.i.d. random variables, local large deviations, Nagaev’s result, Tkachuk and Doney’s results, Diophantine measures and LLT, Breuillard, Mukhin, Shepp, Stone, LLT and Edgeworth expansions, Breuillard and Feller’s LLTs under arithmetical conditions.

The LLT for sums of independent random variables: Prokhorov’s theorem, Richter’s LLTs and large deviations, Maejima’s LLTs with remainder term, LLTs with convergence in variation, Gamkrelidze’s results, Rozanov’s necessary condition and uniform asymptotic distribution, Mitalauskas’ LLT for random variables having stable limit distribution, Azlarov and Gamkrelidze’s counterexamples, structural characteristics, Bernoulli part extraction, Dabrowski and McDonald’s LLT, Giuliano and Weber’s LLT with effective rate, Macht and Wolf’s LLT using Hölder continuity, Röllin and Ross LLTs using Landau–Kolmogorov inequalities, Delbaen, Jacod, Kowalski and Nikeghbali’s recent LLTs under mod-$ \phi$ convergence, Dolgopyat’s recent LLT for sums of independent random vectors satisfying appropriate tightness assumptions, Feller’s LLTs and domains of attraction.

The LLT for ergodic sums: essentially centered around interval expanding maps, results of Kac, Rousseau-Egele, Broise, Calderoni, Campanino and Capocaccia, and more recently Gouëzel and Szewczak.

An expanded and detailed list of applications of the local limit theorem is provided. The last part of the survey is devoted to the more recent study of the almost sure local limit theorem, instilled by Denker and Koch. The inherent second order study, which has its own interest, is much more difficult than for establishing the almost sure central limit theorem. 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, and the independent case, with almost sure convergence of related series.

Our aim in writing this monograph was to survey and make known many interesting results obtained since the sixties. Many of them were obtained in the sixties by the Lithuanian and Russian schools of probability, and are essentially written in Russian, and moreover often published in journals of difficult access. Our intention was to somehow help researchers with this whole coherent body of results and methods in the study of the local limit theorem.