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.

翻译：本文首先研究了具有高斯新息的两个独立AR(1)过程经验相关性的精确分布与渐近分布。随后，我们发展了缩放经验相关性（即经验相关系数乘以数据点数的平方根再乘以归一化常数）分布向标准高斯分布收敛的速度，分别基于Wasserstein距离和Kolmogorov距离。给定$n$个数据点，我们证明了在Wasserstein距离下的收敛速度为$n^{-1/2}$，在Kolmogorov距离下的收敛速度为$n^{-1/2} \sqrt{\ln n}$。接着，我们计算了另外两类AR(1)过程的缩放经验相关性向标准高斯分布的收敛速度：（i）具有相关高斯增量的两个AR(1)过程，以及（ii）由第二Wiener混沌中的白噪声驱动的两个独立AR(1)过程。