Title: 

Yule’s “nonsense correlation”: Moments and density.

Abstract: 

In 1926, G. Udny Yule considered the following problem: given two independent and identically distributed random walks independent from each other, what is the distribution of their empirical correlation coefficient? Yule empirically observed the distribution of this statistic to be heavily dispersed and frequently large in absolute value, leading him to call it “nonsense correlation.'' This unexpected finding led to his formulation of two concrete questions, each of which would remain open for more than ninety years: (i) Find (analytically) the variance of this empirical correlation coefficient and (ii): Find (analytically) the higher order moments and the density of this empirical correlation coefficient. After giving a brief overview of the solution to question (i) in Ernst et al. (The Annals of Statistics, 2017), we turn to the recent work of Ernst et al. (Bernoulli, 2025), which closed question (ii) by explicitly calculating all moments of the empirical correlation coefficient (up to order 16). This leads, for the first time, to an approximation to the density of Yule's nonsense correlation. The methodology of Ernst et al. (2025) further enables explicit calculations of the moments of the empirical correlation coefficient when the two independent Wiener processes are replaced by two correlated Wiener processes, two independent Ornstein-Uhlenbeck processes, and two independent Brownian bridges. We also succeed in proving a Central Limit Theorem for the case of two independent Ornstein-Uhlenbeck processes. This shows that Yule's “nonsense correlation” is indeed not “nonsense” for stochastic processes which admit stationary distributions. The talk concludes with a discussion of some concrete applications of our work to the study of weather and climate extremes. The latter is part of our ongoing collaboration with the U.S. Office of Naval Research (2018-present).

Bio: 

Philip Ernst is Chair (and Full Professor) in Statistics and Royal Society Wolfson Fellow at Imperial College London. He was previously an Assistant Professor (2014-2018), an Associate Professor (2019-2022), and a Full Professor (2022-2023), all at Rice University’s Department of Statistics. His research lies at the interface of applied probability and operations research. His work has been funded by the U.S. Office of Naval Research (ONR), the U.S. Army Research Office (ARO), the National Science Foundation (NSF), The Royal Society, and The British Academy. Ernst is the recipient of numerous international and national research awards, including: a 2026 Institute of Mathematical Statistics (IMS) Medallion Award & Lecture, a 2023 Henri Lebesgue Chair, a 2023 British Academy/Wolfson Fellowship, a 2022 Committee of Presidents of Statistical Societies (COPSS) 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 Institute of Mathematical Statistics (IMS) Tweedie New Researcher Award. Ernst is also highly invested in teaching; he won seven teaching awards in the eight years he was employed at Rice University (including the George R. Brown Prize for Excellence in Teaching, Rice University’s most prestigious teaching award). He currently serves as an associate editor for six journals: Journal of Stochastic Analysis, Journal of the American Statistical Association: Theory and Methods, Mathematics of Operations Research, Statistics and Probability Letters, Stochastics, and The American Statistician. He is also an elected member of IMS Council (2024-2027).