Prof. Yaozhong Hu : Malliavin calculus and convergence in density of some nonlinear Gaussian functionals

Posted by Cheng Ouyang , part of the Statistics and Data Science Seminar.

At

Oct. 23, 2013, 4 p.m.

In

SEO 636

Abstract

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