Philip A. Ernst is Chair in Statistics and Royal Society Wolfson Fellow in the Department of Mathematics, Imperial College London. He earned his PhD in Statistics in 2014 from The Wharton School of the University of Pennsylvania. His research interests include exact distribution theory, mathematical finance, optimal stopping, queueing systems, statistical inference for stochastic processes, and stochastic control. Ernst’s work has been funded by the U.S. Army Research Office (ARO), the U.S. Office of Naval Research (ONR), the National Science Foundation (NSF), The British Academy, and The Royal Society. He is the recipient of numerous international and national research awards, including: a 2026 IMS Medallion Award & Lecture, a 2023 Lebesgue Chair, a 2023 British Academy/Wolfson Fellowship, a 2022 Royal Society Wolfson Fellowship, a 2022 COPSS Leadership Academy (now “Emerging Leader”) Award, the 2020 (inaugural) INFORMS Donald P. Gaver, Jr. Early Career Award for Excellence in Operations Research, a 2018 U.S. Army Research Office (ARO) Young Investigator Award, and the 2018 IMS Tweedie New Researcher Award. He is also a Class of 2025 IMS Fellow. Before joining Imperial College London in 2022, Ernst was Professor of Statistics at Rice University, where he was awarded seven teaching awards in his eight years of employment (including the George R. Brown Prize for Excellence in Teaching: the university’s most prestigious teaching award). He presently serves as Deputy Editor of Stochastics and as Associate Editor for six journals, including Journal of the American Statistical Association, Theory and Methods and Mathematics of Operations Research. This Medallion lecture will be delivered at the 45th Conference on Stochastic Processes and their Applications at Cornell University, Ithaca, USA, June 14–20, 2026.

Yule’s “nonsense correlation”: Moments, density, and tests of independence for non-stationary Gaussian processes

In 1926, G. Udny Yule considered the following problem: given two independent and identically distributed random walks of length n, what is the distribution of their empirical (Pearson) correlation coefficient ρ_n? Yule observed empirically that this statistic ρ_n does not converge to 0, but tends to be frequently far from 0 and heavily dispersed in (−1, 1), leading him to call it “nonsense correlation.”

This led to a formulation of two concrete questions, each of which would remain open for more than 90 years:

(i) Find (analytically) the variance of ρ_n as n tends towards infinity; and

(ii) Find (analytically) the higher order moments and the density of ρ_n as n tends towards infinity.

A straightforward application of the classical Donsker’s theorem shows that the limiting law for ρ_n is the natural empirical correlation ρ of two independent Wiener processes on [0, 1].

In 2017, Ernst, Shepp, and Wyner closed question (i) by calculating Var(ρ) analytically as an explicit double integral, numerically equal approximately to 0.2405. This talk begins where Ernst et al. (2017) leaves off. I shall explain how Ernst, Rogers, and Zhou (2025) succeeded in closing question (ii) by calculating all moments of ρ, explicitly up to order 16, leading, for the first time, to an approximation to the limiting density of Yule’s nonsense correlation.

I will then turn to an investigation of the fluctuations around the convergence of ρ_n to ρ for both standard Brownian motion and long-memory fractional Brownian motion. I will present the key elements of a surprising result of Ernst, Viens, and Yan (2025) by which the fluctuations are of order 1/n rather than 1/Ο−n, and by which the asymptotic distribution is in the so-called second Wiener chaos, whereas its conditional asymptotic law given the data is actually Gaussian. This subtle phenomenon illustrates a conditional central limit theorem (CLT), where the asymptotic variance is data-dependent and there is a non-zero asymptotic mean which is also data dependent. The expressions for these asymptotic first and second moments are explicit, given via bilinear functionals of the data’s two paths, combined with a conditional delta method. The limit in law in the second Wiener chaos can be represented via the same trivariate bilinear functional of the data, and a further trivariate bilinear functional which depends linearly on the data and linearly on an auxiliary pair of Wiener processes independent of the data. The presence of this auxiliary pair causes the limit to be in law and not in a stronger sense; it also explains why the limit can be interpreted as a conditional CLT.

To the best of our knowledge, these results lead to the first statistical test of independence for pairs of non-stationary and non-i.i.d. Gaussian processes, leveraging a diffuse limit in law, even extending to processes with long-range dependent increments.

References

Ernst, PA, Rogers, LCG, and Zhou, Q. (2025) Yule’s “nonsense correlation”: moments and density. Bernoulli, 31(1): 412–431

Ernst, PA, Shepp, LA, and Wyner, AJ. (2017) Yule’s “nonsense correlation” solved! The Annals of Statistics 45(4): 1789-1809.

Ernst, PA, Viens, FG, and Yan, S. (2025) Tests of independence for pairs of paths of non-
stationary Gaussian processes. arXiv: 2510.23563.