The Annals of Probability

Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

Ivan Nourdin, Giovanni Peccati, and Gesine Reinert

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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.

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Ann. Probab., Volume 38, Number 5 (2010), 1947-1985.

First available in Project Euclid: 17 August 2010

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Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus

Central limit theorems chaos homogeneous sums Lindeberg principle Malliavin calculus chi-square limit theorems Stein’s method universality Wiener chaos


Nourdin, Ivan; Peccati, Giovanni; Reinert, Gesine. Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 (2010), no. 5, 1947--1985. doi:10.1214/10-AOP531.

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