Approximate Local Limit Theorems with Effective Rate and Application to Random Walks in Random Scenery

TL;DR

This paper develops a method to derive approximate local limit theorems with explicit error bounds for sums of lattice-valued variables and applies it to random walks in random scenery, improving understanding of their probabilistic behavior.

Contribution

It introduces a Bernoulli part extraction technique to obtain effective local limit theorems with explicit error terms for lattice sums and random walks in random scenery.

Findings

Derived approximate local limit theorems with explicit error bounds.

Recovered classical local limit theorems of Gnedenko and Gamkrelidze.

Established a local limit theorem with effective remainder for random walks in random scenery.

Abstract

We show that the Bernoulli part extraction method can be used to obtain 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. 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.