Rates of uniform convergence of empirical means with mixing processes
Let \({\mathcal X}\) be a set with a \(\sigma\)-algebra \({\mathcal S}\) of subsets of \({\mathcal X}\) and \({\mathcal F}\) a family of \([0,1]\)-valued Borel measurable functions on \(({\mathcal X}, {\mathcal S})\). Let \(\{X_j; -\infty<j <\infty\}\) be a sequence of random variables defined by \(X_j({\mathbf x})=x_j\) \(({\mathbf x}= (\dots, x_{-1},x_0, x_1,x_2,\dots) \in{\mathcal X}^\infty)\). For a shift invariant measure \(P\) define the product measure \(P^*\) on \(({\mathcal X}^\infty,{\mathcal S}^\infty)\) with the same one-dimensional marginals as \(P\). Then, \(\{X_j\}\) is a stationary sequence under \(P\) and is an i.i.d. sequence under \(P^*\). Let \({\mathcal P}\) be a family of shift invariant probability measures \(P\) on \(({\mathcal X}^\infty, {\mathcal S}^\infty)\) and put \({\mathcal P}^* =\{P^*;P\in {\mathcal P}\}\). \(\{X_j\}\) is said to be \(\beta\)-mixing if for \(P\in{\mathcal P}\) \[ \beta(k,P)= \sup\biggl \{\bigl|P(A)- (P^0_{-\infty} \times P_1^\infty) (A)\bigr|;\;A\in\sigma (X_j\leq 0 \text{ or }j\geq k)\biggr\} \to 0 \] where \(P^0_{-\infty}\) is the law of \(\{X_j; -\infty< j\leq 0\}\) and \(P_1^\infty\) is the law of \(\{X_j; 1\leq j<\infty\}\). \({\mathcal P}\) is called a uniformly \(\beta\)-mixing family with mixing rate \(\{\beta (k)\}\) if for all \(P\in{\mathcal P}\), \(\beta(k,P) \leq\beta (k)\downarrow 0\). Set \[ \overline q(m,\varepsilon,P) =\sup_{P\in {\mathcal P}}P\left( \sup_{f\in{\mathcal F}} \left|{1\over m}\sum^m_{i=1} f(X_i)- E_Pf(X_0)\right |> \varepsilon \right) \] and define \(\overline q(m,\varepsilon, {\mathcal P}^*)\) similarly. The authors prove that for a uniformly \(\beta\)-mixing family \({\mathcal P}\) \[ \overline q(m,\varepsilon, {\mathcal P}^*) \to 0\;(\forall \varepsilon>0) \Rightarrow\overline q(m,\varepsilon, {\mathcal P})\to 0\;(\forall \varepsilon>0). \] This result extends the result due to \textit{A. Nobel} and \textit{A. Dembo} [Stat. Probab. Lett. 17, No. 3, 169-172 (1993; Zbl 0776.60042)]. Further, explicit upper bounds on the rate are also derived.
scientific article; zbMATH DE number 1906920