Colloquium Speaker - Dr. Frederi Viens, Rice University

Abstract:

We study the continuous-time version of the empirical correlation coefficient between the paths of two correlated Ornstein-Uhlenbeck processes. Using sharp tools from the analysis on Wiener chaos, we establish the asymptotic normality of the fluctuations of this correlation coefficient around its long-time limit, which is the mathematical correlation coefficient between the two processes. This asymptotic normality is quantified in Kolmogorov distance, which allows us to establish speeds of convergence in the Type-II error for two simple tests of independence of the paths, based on the empirical correlation, and based on its numerator. In this presentation, we explain this scalar setup and how to apply it to testing independence of two observations of solutions to the stochastic heat equation. We explain why this test features excellent asymptotic power properties using merely a small number of the solutions' Fourier modes. We will also discuss how one might apply this type of methodology to attribution questions in observational studies for space-time environmental time series, including combining statistical paleoclimate reconstructions and projections with global circulation models. The mathematical portion of this talk is based on work with Soukaina Douissi and Philip Ernst, in https://arxiv.org/abs/2504.17175 .