Yule's "nonsense correlation" for Gaussian random walks

TL;DR

This paper derives exact formulas for the moments of empirical correlations of Gaussian random walks, revealing their convergence behavior to Wiener processes and quantifying the rate of convergence.

Contribution

It provides explicit formulas for moments of empirical correlations of Gaussian walks and analyzes their convergence to Wiener process correlations.

Findings

Exact second moment formula for empirical correlation of Gaussian walks

Convergence rate in Wasserstein distance is inverse of data points, n^{-1}

Coupling discrete and continuous correlations on a common probability space

Abstract

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 $n^{-1}$.