The Method of Cumulants for the Normal Approximation

TL;DR

This paper surveys the method of cumulants for normal approximation, highlighting its theoretical foundations, bounds, and applications, especially for variables with weaker conditions than exponential moments.

Contribution

It provides a self-contained proof of key lemmas, introduces the Cramér-Petrov series, and discusses methods for cumulant bounds and recent applications.

Findings

Bound on cumulants weaker than Cramér's condition

Connections with heavy-tailed Weibull variables and moderate deviations

Review of cumulant bounding methods and applications

Abstract

The survey is dedicated to a celebrated series of quantitave results, developed by the Lithuanian school of probability, on the normal approximation for a real-valued random variable. The key ingredient is a bound on cumulants of the type $|\kappa_j(X)| \leq j!^{1+\gamma} /\Delta^{j-2}$, which is weaker than Cram\'er's condition of finite exponential moments. We give a self-contained proof of some of the "main lemmas" in a book by Saulis and Statulevi\v{c}ius (1989), and an accessible introduction to the Cram\'er-Petrov series. In addition, we explain relations with heavy-tailed Weibull variables, moderate deviations, and mod-phi convergence. We discuss some methods for bounding cumulants such as summability of mixed cumulants and dependency graphs, and briefly review a few recent applications of the method of cumulants for the normal approximation.