The method of cumulants for the normal approximation

Doering, Hanna; Jansen, Sabine und Schubert, Kristina (2022): The method of cumulants for the normal approximation. In: Probability Surveys, Bd. 19: S. 185-270

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 vertical bar kappa(j)(X)vertical bar <= j!(1+gamma)/Delta(j-2), which is weaker than Cramer's condition of finite exponential moments. We give a self-contained proof of some of the main lemmas in a book by Saulis and StatuleviCius (1989), and an accessible introduction to the Cramer-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.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Informatik
• Themengebiete: 000 Informatik, Informationswissenschaft, allgemeine Werke > 004 Informatik
• ISSN: 1549-5787
• Sprache: Englisch
• DFG: Gefördert durch die Deutsche Forschungsgemeinschaft (DFG) - 318196255