Statistics Seminar

Abstract: Since the empirical work of G. Udny Yule in 1926, it has been known that the classical Pearson correlation coefficient for two independent random walks does not converge to 0 as the number of datapoints $n$ increases to infinity, but rather has a diffuse limiting distribution. Recently, the speaker’s collaborator Philip Ernst calculated the variance of this limit, after the problem remained open for 90 years. This triggered new interest in the mathematics of this correlation coefficient. Ignorance of the aforementioned 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, particularly in environmental observational studies, e.g. for attribution in climate science. With the tools of stochastic analysis we have today, proving that Yule's nonsense correlation has a diffuse limit is a rather straightforward application of the classical Donsker's theorem (invariance principle), where the limiting law is that of a ratio of two quadratic functionals of two Wiener processes on [0.1]. In contrast, when the time series are asymptotically stationary and have moderate memory (like AR(p) processes), the limit is zero in probability.

In this talk, we investigate the fluctuations around both types of convergence. In the case of pairs of stationary time series, we present Gaussian fluctuation results with accompanying Berry-Esseen-type rates of convergence, leading to a straightforward methodology for testing independence, even in finite sample cases. In the case of pairs of random walks, we present elements of a surprising result by which the fluctuations are of order $1/n$ rather than $1/n^{1/2}$, and by which the asymptotic distribution is in the so-called second Wiener chaos, whereas its conditional law given the data is actually Gaussian. This subtle phenomenon illustrates a conditionally central limit theorem (CLT), where the asymptotic variance is data-dependent and there is a non-zero asymptotic mean which is also data dependent, even though the Pearson correlation has a symmetric law. This appears to be a new type of CLT. 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 $1/n$  is not accidentally related to the exotic unconditional convergence in law in the second Wiener chaos, even when the time series are non-Gaussian.

This presentation describes joint work with Philip Ernst (Imperial College London, University of Alabama), Shuo Yan (Imperial College London), Soukaina Douissi (University Cadi Ayyad, Marrakech, Morocco), and John (Tripp) Roberts (Rice University).