Rates of uniform convergence of empirical means with mixing processes

Abstract

It has been shown previously by Nobel and Dembo (Stat. Probab. Lett. 17 (1993) 169) that, if a family of functions F has the property that empirical means based on an i.i.d. process converge uniformly to their values as the number of samples approaches infinity, then F continues to have the same property if the i.i.d. process is replaced by a β-mixing process. In this note, this result is extended to the case where the underlying probability is itself not fixed, but varies over a family of measures. Further, explicit upper bounds are derived on the rate at which the empirical means converge to their true values, when the underlying process is β-mixing. These bounds are less conservative than those derived by Yu (Ann. Probab. 22 (1994) 94).