Maximal Inequalities for Empirical Processes under General Mixing Conditions with an Application to Strong Approximations

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.