Asymptotics of Yule’s nonsense correlation for Ornstein-Uhlenbeck paths: A Wiener chaos approach

In this paper, we study the distribution of so-called “Yule’s nonsense correlation statistic” on a time interval [0,T] for horizon T>0, when T is large, pair (X1,X2) independent Ornstein-Uhlenbeck processes. This statistic by definition equal to: ρ(T):=Y12(T) Y11(T) Y22(T), where 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 same drift parameter θ>0. also asymptotic law discrete-type version ρ(T), Yij(T) above replaced their Riemann-sum discretizations. case, conditions provided how discretization (in-fill) step relates to long T. establish identical normal asymptotics standardized ρ(T) its discrete-data version. The variance ρ(T)T1∕2 θ−1. speeds convergence in Kolmogorov distance, which Berry-Esséen-type (constant*T−1∕2) except lnT factor. Our method use properties Wiener-chaos variables, since discrete comprised ratios involving three such 2nd Wiener chaos. methodology accesses distance thanks relation stems from connection between Malliavin calculus Stein’s space.