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

Abstract

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 $(X_1,X_2)$ of independent Ornstein-Uhlenbeck processes. This statistic is by definition equal to:

$$\mathit{\rho }(T):=\frac{Y_{12}(T)}{\sqrt{Y_{11}(T)}\sqrt{Y_{22}(T)}},$$

where the random variables $Y_{ij}(T)$, $i,j=1,2$ are defined as

$$Y_{ij}(T):={\int }_0^T{X}_i(u)X_j(u)du-T{\overline{X}}_i\overline{X_j},{\overline{X}}_i:=\frac{1}{T}{\int }_0^T{X}_i(u)du.$$

We assume $X_1$ and $X_2$ have the same drift parameter $\mathit{\theta }>0$. We also study the asymptotic law of a discrete-type version of $\mathit{\rho }(T)$, where $Y_{ij}(T)$ 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 $\mathit{\rho }(T)$ and its discrete-data version. The asymptotic variance of $\mathit{\rho }(T)T^{1∕2}$ is ${\mathit{\theta }}^{-1}$. 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 $\mathit{\rho }(T)$ 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.