Probability Seminar: Yaozhong Hu

The classical central limit theorem is one of the most important theorem in probability theory.  The theorem states that if X_1, \cdots , X_n are independent identically distributed random variables and if F_n is the difference between the sample mean and the mean of the random variables properly normalized, then F_n converges to a normal distribution in distribution.  Recent results extend this results to other random variables for example given by Wiener chaos (multiple It\^o-Wiener integrals). In this talk, we shall obtain some conditions on F_n such that the distributions of the random variables F_n have densities f_n(x)  with respect to Lebesgue  measure and f_n(x) converges to the normal density \phi(x)=\frac{1}{\sqrt{2\pi}}e^{-|x|^2/2}.

The tool that we use is the Malliavin calculus and a brief introduction will also be given.

This is an ongoing joint work with Fei Lu and David Nualart.