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

This paper begins with a study of both the exact distribution and the asymptotic distribution of the empirical correlation of two independent AR(1) processes with Gaussian innovations. We proceed to develop rates of convergence for the distribution of the scaled empirical correlation %(i.e. the empirical correlation times the square root of the number of data points times a normalized constant) to the standard Gaussian distribution in both Wasserstein distance and in 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 then compute rates of convergence of the scaled empirical correlation to the standard Gaussian distribution for two additional classes of AR(1) processes: (i) two AR(1) processes with correlated Gaussian increments and (ii) two independent AR(1) processes driven by white noise in the second Wiener chaos.