Entropy conditions for \(L_{r}\)-convergence of empirical processes (Q1016138)

Let \(P_n\) be the empirical measure for a sample of \(n\) independent random elements with joint distribution \(P\). Furthermore, let \({\mathcal F}\) be a class of real-valued (bounded) functions defined on the sample space. Then it is known that \(\sup_{f\in {\mathcal F}}|P_nf - Pf| \to 0\) almost surely provided that \({\mathcal F}\) admits a finite entropy integral. In the present paper the appropriate entropy is discussed to ensure convergence of \(\int_{{\mathcal F}}|P_nf-Pf|^r d\mu(f)\) to zero, where \(\mu\) is a given measure on \({\mathcal F}\).