Normal Approximation on a Finite Wiener Chaos
The purpose of this note is to survey some recent developments in the applications of Malliavin calculus combined with Stein’s method to derive central limit theorems for random variables on a finite sum of Wiener chaos. Starting from the fourth moment theorem by Nualart and Peccati , we will discuss several related topics such as conditions for the convergence in total variation, absolute continuity of probability laws and uniform convergence of densities under suitable non degeneracy assumptions. The fact that the random variables belong to a fixed Wiener chaos (or to a finite sum of Wiener chaos) will play a fundamental role in the results.
KeywordsMalliavin calculus Wiener chaos Total variation distance Stein’s method
2010 Mathematics Subject Classification60G15 60H07 60H10 65C30
We would like to thank an anonymous referee for carefully reading this manuscript and providing useful remarks.
- 9.Y. Hu, D. Nualart, S. Tindel, F. Xu, Density convergence in the Breuer-Major theorem for nonlinear Gaussian stationary sequences. Bernoulli (To appear)Google Scholar
- 11.P. Malliavin, Stochastic calculus of variation and hypoelliptic operators, in Proceedings of the International Symposium on Stochastic Differential Equations, Kyoto, 1976, pp. 195–263, (Wiley, New York, 1978)Google Scholar
- 13.I. Nourdin, D. Nualart, Fisher information and the fourth moment theorem. Ann. Inst. H. Poincaré Probab. Statist. (To appear)Google Scholar
- 24.D. Nualart, C.A. Tudor, The determinant of the iterated Malliavin matrix and the density of a couple of multiple integrals (Preprint)Google Scholar
- 25.G. Peccati, C. A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals in Séminaire de Probabilités XXXVIII, 247–262, Lecture Notes in Math., 1857, (Springer, Berlin, 2005)Google Scholar
- 26.R. Shimizu, On Fisher’s amount of information for location family in Proceedings of A Modern Course on Statistical Distributions in Scientific Work, pp. 305–312, (Springer, Berlin, 1975)Google Scholar