Yule's "nonsense correlation" for Gaussian random walks

The purpose of this article 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 \cite{ernst2019distribution} and investigated previously in \cite{ernst2017yule}.