Some limit theorems for the empirical process indexed by functions
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.
scientific article; zbMATH DE number 3942612
scientific article