期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2012
卷号：IX
期号：2
页码：473-500
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：Let {Fn : n > 1} be a normalized sequence of random variables insome fixed Wiener chaos associated with a general Gaussian field, and assume thatE[F4n] ! E[N4] = 3, where N is a standard Gaussian random variable. Our mainresult is the following general bound: there exist two finite constants c,C > 0such that, for n sufficiently large, c × max(|E[F3n]|,E[F4n] − 3) 6 d(Fn,N) 6C × max(|E[F3n]|,E[F4n] − 3), where d(Fn,N) = sup |E[h(Fn)] − E[h(N)]|, andh runs over the class of all real functions with a second derivative bounded by 1.This shows that the deterministic sequence max(|E[F3n]|,E[F4n] − 3), n > 1, completelycharacterizes the rate of convergence (with respect to smooth distances) inCLTs involving chaotic random variables. These results are used to determine optimalrates of convergence in the Breuer-Major central limit theorem, with specificemphasis on fractional Gaussian noise.
