Résumé : In 1926 Statistician G. Udny Yule showed that for two independent standard random walks, the empirical correlation coefficient (Pearson's correlation) does not converge to 0, but rather appears to converge in distribution to a diffuse law supported by the entire interval (-1,1). This phenomenon, which has since been recognized for many highly non-stationary time series, is in sharp contrast with the classical result for two sequences of i.i.d. data, by which the same correlation converges to zero, a phenomenon which also extends to many stationary time series. Still, ignorance of this empirical fact for random walks and other non-stationary time series, known today as Yule's nonsense correlation, has lead practitioners to make dramatically ill-informed assertions about statistical associations. This improper use of methodology has occurred in recent times, particularly in environmental observational studies, e.g. for attribution in climate science. The mathematics behind the basic premise of Yule's nonsense correlation are a rather straightforward application of the classical Donsker's theorem; the Pearson correlation ρn of two random walks of length n converges in distribution to the law of a random variable ρ written explicitly as the ratio of two quadratic functionals of two Wiener processes on [0.1]. In this talk, we investigate the fluctuations around this convergence. We present elements of a new result by which n(ρ − ρn) has an asymptotic distribution in the so-called second Wiener chaos, whose characteristics are partly exogenous to the original data, as one would expect for a standard central limit theorem, and are partly conditional on the data. We will discuss the implications of this discovery in practical testing for independence and for attribution in environmental time series. We conjecture that the fluctuation scale, of order 1/n rather than 1/n1/2, is not accidentally related to the exotic convergence in law in the second Wiener haos.

Keywords : nonsense correlation; gaussian random walks; Wiener processes; Ornseint-Uhlenbeck processes; Wasserstein distance; testing path independence