Classical and Almost Sure Local Limit Theorems

Abstract

In this monograph, 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, 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 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 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 first to share the passion resulting from the study for this very special limit theorem, second 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. In doing so, our feeling was to somehow help with this whole coherent body of results and methods, researchers in the study of the local limit theorem, at least it is our hope.

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

“…, d − 1, d ≥ 2, and further S n /σ n asymptotically is normal, but the local limit theorem does not hold. By Rozanov's theorem, the asymptotic uniform distribution property is a necessary condition for the validity of the local limit theorem, see [9]. This is recently improved in [11].…”