Approximate local limit theorems with effective rate and application to random walks in random scenery

Abstract

We show that the Bernoulli part extraction method can be used to obtain approxi- mate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery. 1. Introduction and Main Result. The extraction method of the Bernoulli part of a (lattice valued) random variable was devel- oped by McDonald in (13),(14),(4) for proving local limit theorems in presence of the central limit theorem. Twenty years before McDonald's work, Kolmogorov (11) initiated a similar approach in the study of Levy's concentration function, and is the first having explored this direction. For details and clarifications, we refer to the recent paper by Aizenmann, Germinet, Klein and Warzel (1), where this idea is also developed for general random variables and applications are given. That method allows to transfer results which are available for systems of Bernoulli random variables to systems of arbitrary random variables. It is based on a probabilistic device, and is proved to be an efficient alternative to the characteristic functions method. Kolmogorov wrotes to this effect in his 1958's paper (11) p.29: ...Il semble cependant que nous restons toujours dans une periode ou la competition de ces deux directions (characteristic functions or direct methods from the calculus of probability) conduit aux resultats les plus feconds .... We believe that Kolmogorov's comment is still topical. The main object of this article is to show that this approach can be used to obtain, in a rather simple way, approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. The approximate form we obtain expresses quite simply, and is thereby very handable. Further, it is precise enough to contain Gnedenko and Gamkrelidze local limit theorems (1.2). Before stating the main results and in view of comparing results, it is necessary to recall and discuss some classical facts and briefly describe the background of this problem. Let ˜