Rates of uniform convergence of empirical means with mixing processes

It has been shown previously by Nobel and Dembo (Stat. Probab. Lett. 17 (1993) 169) that, if a family of functions 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 continues to have the same property if the i.i.d. process is replaced by a [beta]-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 [beta]-mixing. These bounds are less conservative than those derived by Yu (Ann. Probab. 22 (1994) 94).