Esscher Transform and the Central Limit Theorem

TL;DR

This paper investigates the use of Esscher transforms in high-dimensional spaces to understand conditions for normal approximation of sums of i.i.d. vectors, extending previous one-dimensional results using Rényi divergence.

Contribution

It extends the application of Esscher transforms and Rényi divergence to high-dimensional normal approximation, providing new necessary and sufficient conditions.

Findings

Established conditions for normal approximation in high dimensions

Extended one-dimensional results to multi-dimensional settings

Utilized Rényi divergence of infinite order for analysis

Abstract

The paper is devoted to the investigation of Esscher's transform on high dimensional Euclidean spaces in the light of its application to the central limit theorem. With this tool, we explore necessary and sufficient conditions of normal approximation for normalized sums of i.i.d. random vectors in terms of the R\'enyi divergence of infinite order, extending recent one dimensional results.