2021/08/05 by Soukaina Douissi, Douissi, Soukaina, Frédéri Viens +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2108.02857
openalex publication_date 2021/08/05 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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):=\fracY12(T)√Y11(T)√Y22(T), where the random variables Yij(T), i,j=1,2 are defined as Yij(T):=∫0TXi(u)Xj(u)du-TXiXj, Xi:=(1)/(T)∫0TXi(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 ln T 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.