Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

Ivan Nourdin, Giovanni Peccati, and Gesine Reinert

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

Abstract

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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Primary Subjects: 60F05, 60F17, 60G15, 60H07

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

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1282053777
Digital Object Identifier: doi:10.1214/10-AOP531
Zentralblatt MATH identifier: 05793427
Mathematical Reviews number (MathSciNet): MR2722791

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