Third Year Candidacy Presentation: Gaussian Approximations and Bootstrap Methods in High-Dimensional Time Series

Abstract: High-dimensional time series analysis has become increasingly critical in statistics and econometrics, propelled by the growing availability of large-scale, temporally correlated datasets. Traditional statistical methods encounter significant obstacles in this context, primarily related to complex dependency structures, inference accuracy, and computational challenges due to high dimensionality. Recent developments involving Gaussian approximations, bootstrap methodologies, and sparse modeling have substantially advanced the analysis and inference techniques in the time domain. Simultaneously, frequency-domain approaches have gained prominence by effectively capturing intricate dependencies among multiple components of high-dimensional time series. A central challenge in this framework is accurately estimating spectral density matrices and their inverses, essential for quantifying spectral coherence and partial coherence. Contemporary research has introduced regularized estimators, established non-asymptotic error bounds, and extended methods to accommodate nonstationary conditions. In this talk, I will provide an overview of these recent methodological advancements in both the time and frequency domains. Additionally, I will discuss potential pathways for further development and their implications for future research in high-dimensional time series analysis.

Advisors: Soumen Lahiri and Todd Kuffner