期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2019
卷号：XVI
期号：1
页码：633-664
DOI：10.30757/ALEA.v16-23
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We study rates of convergence in central limit theorems for the partialsum of squares of general Gaussian sequences, using tools from analysis on Wienerspace. No assumption of stationarity, asymptotically or otherwise, is made. Themain theoretical tool is the so-called Optimal Fourth Moment Theorem (Nourdinand Peccati, 2015), which provides a sharp quantitative estimate of the total variationdistance on Wiener chaos to the normal law. The only assumptions madeon the sequence are the existence of an asymptotic variance, that a least-squarestypeestimator for this variance parameter has a bias and a variance which can becontrolled, and that the sequence’s auto-correlation function, which may exhibitlong memory, has a no-worse memory than that of fractional Brownian motionwith Hurst parameter H < 3=4. Our main result is explicit, exhibiting the tradeoffbetween bias, variance, and memory. We apply our result to study drift parameterestimation problems for subfractional Ornstein-Uhlenbeck and bifractionalOrnstein-Uhlenbeck processes with fixed-time-step observations. These are processeswhich fail to be stationary or self-similar, but for which detailed calculationsresult in explicit formulas for the estimators’ asymptotic normality.
