The purpose of this paper is to provide an exact formula for the second moment of the empirical correlation of two independent Gaussian random walks as well as implicit formulas for higher moments. The proofs are based on a symbolically tractable integro-differential representation formula for the moments of any order in a class of empirical correlations, first established by Ernst et al. (2019) and investigated previously in Ernst et al. (2017). We also provide rates of convergence of the empirical correlation of two independent Gaussian random walks to the empirical correlation of two independent Wiener processes, by exploiting the explicit nature of the computations used for the moments. At the level of distributions, in Wasserstein distance, the convergence rate is the inverse $n^{-1}$ of the number of data points $n$. This holds because we represent and couple the discrete and continuous correlations on a common probability space, where we establish convergence in $L^1$ at the rate $n^{-1}$.

翻译：本文的目的是为两个独立的高斯随机行走以及更高时刻隐含公式的经验相关性的第二个时刻提供一个精确的公式。 证明基于一个象征性的、可移动的异种分布式代表公式,该公式用于某类经验相关性中任何时刻的任何顺序,最初由恩斯特等人(2019年)确立,以前在恩斯特等人(2017年)中调查过。我们还提供了两个独立的高斯随机行走的经验相关性与两个独立的维纳进程的经验相关性的趋同率,方法是利用用于这些时刻的计算的明确性质。 在分布水平上,在瓦塞斯坦距离,趋同率是数据点数的逆值 $ {-1} 。 这是因为我们代表共同概率空间的离散和连续的关联率,我们在该空间以1美元和1美元的比例建立趋同。