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 |κj(X)|≤j!1+γ∕Δj−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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
