Wasserstein and Convex Gaussian Approximations for Non-stationary Time Series of Diverging Dimensionality, with Applications

Zhou Zhou obtained his Ph.D. in Statistics from the University of Chicago in 2009. He is currently a Full Professor at the Department of Statistical Sciences, University of Toronto. Zhou's major research interests lie in complex time series analysis, non- and semi-parametric inference, time-frequency analysis, change point analysis, and functional and longitudinal data analysis. Zhou received the NSERC Discovery Accelerator Award in 2021 and the CRM-SSC Prize in 2023.

In high-dimensional time series analysis, Gaussian approximation (GA) schemes under various distance measures or on various collections of subsets of the Euclidean space play a fundamental role in a wide range of statistical inference problems. To date, most GA results for high-dimensional time series are established on hyper-rectangles and their equivalence. In this talk, by considering the 2-Wasserstein distance and the collection of all convex sets, we establish general GA theory for a very wide class of high-dimensional non-stationary (HDNS) time series, broadening the range of problems that can be addressed in HDNS time series analysis. For HDNS time series of sufficiently weak dependence and light tail, the GA rates established in this paper are either nearly optimal with respect to the dimensionality and time series length, or they are nearly identical to the corresponding best-known GA rates established for independent data. A multiplier bootstrap procedure is utilized and theoretically justified to implement our GA theory. We demonstrate our theory and method by considering joint simultaneous confidence band construction for time series scalar-on-function linear regression.