Some limit theorems for the empirical process indexed by functions (Q1072218)

Let \(G_ n\), \(n\geq 1\), be a sequence of classes of real-valued measurable functions defined on a probability space (S,\({\mathcal S},P)\). Under weak metric entropy conditions on \(G_ n\), \(n\geq 1\), and under growth conditions on \(\sup \{\| g\|_{\infty}:\) \(g\in G_ n\}\) and \(\sup \{\| g\|_ 2:\) \(g\in G_ n\}\), we show that there are non-zero numerical constants \(C_ 1\) and \(C_ 2\) such that \[ C_ 1\leq \overline{\lim}_{n\to \infty}\sup_{g\in G_ n}| \alpha (n)\int g(dP_ n-dP)| \leq C_ 2<\infty \quad a.s.\quad (P^{{\mathbb{N}}}), \] where \(\alpha\) (n) is a non-decreasing function of n related to the metric entropy of \(G_ n\). A few applications of this general result are considered: we obtain a.s. rates of uniform convergence for the empirical process indexed by intervals as well as a.s. rates of uniform convergence for the empirical characteristic function over expanding intervals.