The Method of Cumulants for the Normal Approximation

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 |κ j (X)| ≤ j! 1+γ /∆ j−2 , which is weaker than Cramér's condition of finite exponential moments. We give a self-contained proof of some of the "main lemmas" in a book by Saulis and Statulevičius (1989), and an accessible introduction to the Cramér-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.

The $β$-Delaunay tessellation III: Kendall's problem and limit theorems in high dimensions

“…The proof of Theorem 4.4 relies on the following lemma, which summarizes some of the main finings of the method of cumulants for normal approximation. We refer the reader to the recent survey article [8] and the many references provided therein.…”