Exact and asymptotic distribution theory for the empirical correlation of two AR(1) processes with Gaussian increments

Abstract

This paper begins with a study of the exact distribution of the empirical correlation of two independent AR(1) processes with Gaussian increments. We proceed to develop rates of convergence for the distribution of the scaled empirical correlation to the standard Gaussian distribution in both Wasserstein distance and Kolmogorov distance. Given $n$ data points, we prove the convergence rate in Wasserstein distance is $n^{-1/2}$ and the convergence rate in Kolmogorov distance is $n^{-1/2} \sqrt{\ln n}$. We conclude by extending these results to two AR(1) processes with correlated Gaussian increments.