文章基本信息

标题：Asymptotics of Yule’s nonsense correlation for Ornstein-Uhlenbeck paths: A Wiener chaos approach
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作者：Soukaina Douissi ; Khalifa Es-Sebaiy ; Frederi Viens 等
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2022
卷号：16
期号：1
页码：3176-3211
DOI：10.1214/22-EJS2021
语种：English
出版社：Institute of Mathematical Statistics
摘要：In this paper, we study the distribution of the so-called “Yule’s nonsense correlation statistic” on a time interval [0,T] for a time horizon T>0, when T is large, for a pair (X1,X2) of independent Ornstein-Uhlenbeck processes. This statistic is by definition equal to: ρ(T):=Y12(T) Y11(T) Y22(T), where the random variables Yij(T), i,j=1,2 are defined as Yij(T):= ∫0TX i(u)Xj(u)du−TX¯iXj¯, X¯i:=1 T∫0TX i(u)du. We assume X1 and X2 have the same drift parameter θ>0. We also study the asymptotic law of a discrete-type version of ρ(T), where Yij(T) above are replaced by their Riemann-sum discretizations. In this case, conditions are provided for how the discretization (in-fill) step relates to the long horizon T. We establish identical normal asymptotics for standardized ρ(T) and its discrete-data version. The asymptotic variance of ρ(T)T1∕2 is θ−1. We also establish speeds of convergence in the Kolmogorov distance, which are of Berry-Esséen-type (constant*T−1∕2) except for a lnT factor. Our method is to use the properties of Wiener-chaos variables, since ρ(T) and its discrete version are comprised of ratios involving three such variables in the 2nd Wiener chaos. This methodology accesses the Kolmogorov distance thanks to a relation which stems from the connection between the Malliavin calculus and Stein’s method on Wiener space.
关键词：60F05;60G15;Ornstein-Uhlenbeck paths;Wiener chaos approach;Yule’s nonsense correlation