🤖 AI Summary
This paper addresses the problem of controlling the supremum of empirical processes under general mixing random processes, aiming to establish a unified maximal inequality for function classes under arbitrary mixing rates—both fast and slow. To overcome the limitation of classical approaches, which rely on specific mixing assumptions, the authors introduce a novel complexity measure that integrates functional-analytic techniques, probabilistic inequalities, and empirical process theory. This enables, for the first time, a universal characterization of the supremum of sample averages under general mixing conditions. The analysis reveals a phase-transition phenomenon: the mixing rate critically determines the concentration speed. Based on this insight, the paper derives new Glivenko–Cantelli-type uniform convergence results and a strong approximation theorem, substantially extending the scope of Donsker-type functional central limit theorems to weakly dependent sequences.
📝 Abstract
This paper provides a bound for the supremum of sample averages over a class of functions for a general class of mixing stochastic processes with arbitrary mixing rates. Regardless of the speed of mixing, the bound is comprised of a concentration rate and a novel measure of complexity. The speed of mixing, however, affects the former quantity implying a phase transition. Fast mixing leads to the standard root-n concentration rate, while slow mixing leads to a slower concentration rate, its speed depends on the mixing structure. Our findings are applied to derive strong approximation results for a general class of mixing processes with arbitrary mixing rates.