Rates of Convergence for Empirical Processes of Stationary Mixing Sequences

Classical empirical process theory for Vapnik-Cervonenkis classes deals mainly with sequences of independent variables. This paper extends the theory to stationary sequences of dependent variables. It establishes rates of convergence for β-mixing and φ-mixing empirical processes indexed by classes of functions. The method of proof depends on a coupling of the dependent sequence with sequences of independent blocks, to which the classical theory can be applied. A uniform O(n-s/(1+s)) rate of convergence over V-C classes is established for sequences whose mixing coefficients decay slightly faster than O(n⁻ˢ).