π€ AI Summary
This work addresses the self-normalized concentration problem for multivariate vector-valued stochastic processes under non-sub-Gaussian (including heavy-tailed) settingsβa gap left by existing scalar concentration theories, which do not extend naturally to high dimensions. We propose a unified analytical framework based on generalized sub-Ο tail conditions and establish, for the first time, time-uniform self-normalized concentration inequalities for vector-valued processes. Our analysis yields a sharp (constant-optimal) upper law of the iterated logarithm and develops multivariate empirical Bernstein inequalities. Methodologically, we integrate martingale theory, generalized cumulant generating function techniques, and vector-valued stochastic process analysis. The resulting confidence regions require no prior knowledge of variance, enabling robust and tight inference in linear regression, autoregressive modeling, and bounded-mean estimation. Empirically, our approach significantly strengthens convergence guarantees and estimation robustness under heavy-tailed noise.
π Abstract
Self-normalized processes arise naturally in many learning-related tasks. While self-normalized concentration has been extensively studied for scalar-valued processes, there are few results for multidimensional processes outside of the sub-Gaussian setting. In this work, we construct a general, self-normalized inequality for multivariate processes that satisfy a simple yet broad sub-$psi$ tail condition, which generalizes assumptions based on cumulant generating functions. From this general inequality, we derive an upper law of the iterated logarithm for sub-$psi$ vector-valued processes, which is tight up to small constants. We show how our inequality can be leveraged to derive a variety of novel, self-normalized concentration inequalities under both light and heavy-tailed observations. Further, we provide applications in prototypical statistical tasks, such as parameter estimation in online linear regression, autoregressive modeling, and bounded mean estimation via a new (multivariate) empirical Bernstein concentration inequality.
