We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.
Publié le : 2010-09-15
Classification:
Central limit theorems,
chaos,
homogeneous sums,
Lindeberg principle,
Malliavin calculus,
chi-square limit theorems,
Stein’s method,
universality,
Wiener chaos,
60F05,
60F17,
60G15,
60H07
@article{1282053777,
author = {Nourdin, Ivan and Peccati, Giovanni and Reinert, Gesine},
title = {Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos},
journal = {Ann. Probab.},
volume = {38},
number = {1},
year = {2010},
pages = { 1947-1985},
language = {en},
url = {http://dml.mathdoc.fr/item/1282053777}
}
Nourdin, Ivan; Peccati, Giovanni; Reinert, Gesine. Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab., Tome 38 (2010) no. 1, pp. 1947-1985. http://gdmltest.u-ga.fr/item/1282053777/