Local Limit Theorems in some Random models from Number Theory

TL;DR

This paper investigates local limit theorems for weighted sums of Bernoulli variables in number theory models, proving new results and demonstrating limitations of standard forms in estimating probabilities for infinite sets.

Contribution

It introduces new local limit theorems for Bernoulli sums in arithmetical models and explores their applications and limitations in number theory.

Findings

Proved new local limit theorems using characteristic functions.

Applied almost sure local limit theorem to additive number theory problems.

Showed standard local limit theorem may not be sharp for infinite sets.

Abstract

We study the local limit theorem for weighted sums of Bernoulli variables. We show on examples that this is an important question in the general theory of the local limit theorem, and which turns up to be not well explored. The examples we consider arise from standard random models used in arithmetical number theory. We next use the characteristic function method to prove new local limit theorems for weighted sums of Bernoulli variables. Further, we give an application of the almost sure local limit theorem to a representation problem in additive number theory due to Burr, using an appropriate random model. We also give a simple example showing that the local limit theorem, in its standard form, fails to be sharp enough for estimating the probability $P\{S_n\in E\}$ for infinite sets of integers $E$, already in the simple case where $S_n$ is a sum of $n$ independent standard Bernoulli random variables and $E$ an arithmetic progression.