A conditional strong large deviation result and a functional central limit theorem for the rate function

TL;DR

This paper investigates the large deviation properties and fluctuation behavior of sums of independent random variables, establishing a conditional strong large deviation result and a functional central limit theorem for the rate function.

Contribution

It introduces new results on the large deviation behavior and fluctuation analysis of sums involving independent sequences, extending understanding of rate functions.

Findings

Established a conditional strong large deviation principle.

Proved a functional central limit theorem for the rate function.

Provided insights into the fluctuation behavior of sums of independent variables.

Abstract

We study the large deviation behaviour of $S_n=\sum_{j=1}^n W_jZ_j$, where $(W_j)_{j \in \mathbb N}$ and $(Z_j)_{j \in \mathbb N}$ are sequences of real-valued, independent and identically distributed random variables satisfying certain moment conditions, independent of each other. More precisely, we prove a conditional strong large deviation result and describe the fluctuations of the random rate function through a functional central limit theorem.