A note on uniform laws of averages for dependent processes (Q1801875)

Let \({\mathcal F}\) be a class of real-valued functions defined on a measurable space \(({\mathcal X},{\mathcal S})\) such that \[ \sup_{f\in{\mathcal F}}\left|{1\over n}\sum^{n-1}_{i=0}f(X_ i)-Ef(X_ 0)\right|\to 0 \tag{*} \] almost surely for some \({\mathcal X}\)-valued i.i.d. sequence \(\{X_ n, n\geq 0\}\) with distribution function (d.f.) \(F\). Under certain regularity conditions on \({\mathcal F}\), it is shown that \((*)\) holds for every stationary, absolutely regular stochastic sequence \(\{X_ n,-\infty<n<\infty\}\) whose marginal d.f. is \(F\).