A conditional Berry-Esseen bound and a conditional large deviation result without Laplace transform. Application to hashing with linear probing

TL;DR

This paper establishes a Berry-Esseen bound and a large deviation principle for sums of i.i.d. variables conditioned on other sums, without relying on Laplace transforms, with applications to hashing algorithms.

Contribution

It introduces a novel approach to large deviations and Berry-Esseen bounds without Laplace transforms, applicable to combinatorial models like hashing with linear probing.

Findings

Proved a Berry-Esseen bound in a general setting

Established a large deviation result without Laplace transform

Applied results to hashing with linear probing

Abstract

\noindent We study the asymptotic behavior of a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables. We prove a Berry-Esseen bound in a general setting and a large deviation result when the Laplace transform of the underlying distribution is not defined in a neighborhood of zero. Then we present several combinatorial applications. In particular, we prove a large deviation result for the model of hashing with linear probing.