Normal approximations with Malliavin calculus. From Stein's method to universality (Q2891288)

The book explores different aspects of normal approximations using techniques of Malliavin calculus. It collects and thoroughly explains this modern view on a very classic field of probability theory.NEWLINENEWLINEThe monograph is largely self contained and easily readable, requiring the reader only to have a basic knowledge of probability theory and Gaussian random variables. It provides a comprehensive introduction to Stein's method and the Malliavin calculus of variations, accompanied by many instructive examples and exercises. In addition, supplementary content such as useful results from functional analysis, Gaussian techniques, a chapter on probability distances and a review of fractional Brownian motion is given in the appendix.NEWLINENEWLINE The topic of this book, normal approximations, studies the convergence of two or more random objects, involving one or more normal (i.e., Gaussian) distributions. More precisely, the text is mainly concerned with the quantification of the distance between their laws.NEWLINENEWLINEA powerful tool to obtain such distance estimates is the celebrated Stein's method. The method gives rise to a collection of probabilistic techniques that allow to link the distance of distributions to differential operators. For instance, these techniques provide a beautiful analytic way to derive the classic Berry-Esseen bound in the context of the standard central limit theorem.NEWLINENEWLINEOn the other hand, the mighty Malliavin calculus of variations has been developed independently as a differential calculus on a Gaussian space, originally designed to investigate the smoothness of probability laws.NEWLINENEWLINEThe connection between the two differential calculi has been established by the authors over the last years and is the core of this monograph.NEWLINENEWLINEThe method applies to functionals of Gaussian random fields such as (fractional) Brownian motion and Gaussian chaos or more general functionals. To this aim, it contains a broad introduction to both of these very active fields.NEWLINENEWLINEWith these tools at hand, the book examines classic results such as the Lindeberg and Breuer-Major theorem, the method of moments and cumulants and Edgeworth expansions from a new perspective.NEWLINENEWLINEFurthermore, many recent developments and applications, optimality and universality results are studied in detail.