Group Characters and Malliavin Calculus

Abstract

In Chap. 14 two applications of Stein’s method are considered, each of which go well beyond the confines of the method’s originally intended uses; the approximation of the distribution of characters of elements chosen uniformly from compact Lie groups, and of random variables in a fixed Wiener chaos of Brownian motion, using the tools of Malliavin calculus. Regarding the first application, the chapter considers the study of random characters as a type of a generalization to abstract groups of the study of traces of random matrices, and hence of the combinatorial central limit theorem. Bounds to the normal are given for characters of random elements from the orthogonal group, the special orthogonal group, and the unitary symplectic group. As for the second application, joining Stein’s method to Malliavin calculus shows that the underlying fundamentals of Stein’s method, in particular the basic characterization of the normal which can be shown by integration by parts, can be extended, with great benefit, to abstract Wiener spaces. Using the methods developed in this chapter, bounds in the total variation metric to the normal are given for functionals in the qth Weiner chaos of Brownian motion for general q≥2.