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标题：A third-moment theorem and precise asymptotics for variations of stationary Gaussian sequences
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作者：Léo Neufcourt ; Frederi G. Viens
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2016
卷号：XIII
页码：239-264
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：In two new papers Bierm´e et al. (2012) and Nourdin and Peccati (2015),sharp general quantitative bounds are given to complement the well-known fourthmoment theorem of Nualart and Peccati, by which a sequence in a fixed Wienerchaos converges to a normal law if and only if its fourth cumulant converges to 0.The bounds show that the speed of convergence is precisely of order the maximumof the fourth cumulant and the absolute value of the third moment (cumulant).Specializing to the case of normalized centered quadratic variations for stationaryGaussian sequences, we show that a third moment theorem holds: convergenceoccurs if and only if the sequence’s third moments tend to 0. This is proved forsequences with general decreasing covariance, by using the result of Nourdin andPeccati (2015), and finding the exact speed of convergence to 0 of the quadraticvariation’s third and fourth cumulants. Nourdin and Peccati (2015) also allowsus to derive quantitative estimates for the speeds of convergence in a class of logmodulatedcovariance structures, which puts in perspective the notion of criticalHurst parameter when studying the convergence of fractional Brownian motion’squadratic variation. We also study the speed of convergence when the limit is notGaussian but rather a second-Wiener-chaos law. Using a log-modulated class ofspectral densities, we recover a classical result of Dobrushin-Major/Taqqu wherebythe limit is a Rosenblatt law, and we provide new convergence speeds. The conclusionin this case is that the price to pay to obtain a Rosenblatt limit despite aslowly varying modulation is a very slow convergence speed, roughly of the sameorder as the modulation.