Local limit theorems in some random models from number theory. Issue 6 (1st November 2016)

• Record Type: Journal Article
• Authors: Giuliano, Rita; Weber, Michel
• Is Part Of: Stochastic analysis and applications. Volume 34:Issue 6(2016)
• Page Start: 941
• Page End: 960
• Publication Date: 2016-11-01
• Subjects: independent random variables -- lattice distributed -- Bernoulli part -- local limit theorem -- almost sure local limit theorem -- effective remainder -- random walk in random scenery; Primary: 60F15 -- 60G50 -- Secondary: 60F05; Stochastic analysis -- Periodicals; 519.2205
• DOI: 10.1080/07362994.2016.1191994
• Languages: English
• ISSNs: 0736-2994

Abstract

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 for infinite sets of integers E, already in the simple case where Sn is a sum of n independent standard Bernoulli random variables and E an arithmetic progression.