  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 $n^{-1}$.
