Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

Details Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

2009-04-07

arxiv.org/abs/0904.1153v2

Annals of Probability 2010, Vol. 38, No. 5, 1947-1985

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP531 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP531

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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