Convergence rates for function classes with applications to the empirical characteristic function (Q1089990)

Let \(G_ n\), \(n\geq 1\), be a sequence of classes of real-valued measurable functions defined on a probability space (S,\({\mathcal S},P)\). Using weak metric entropy integral conditions on \(G_ n\), \(n\geq 1\), as well as growth conditions on the maximal \(L^ 2\) norm \(V(n):=\sup \{\int g^ 2dP:\) \(g\in G_ n\}\) and the maximal sup-norm \(B(n):=\sup \{\| g\| _{\infty}:\) \(g\in G_ n\}\), we provide a.s. uniform rates of convergence for the normalized empirical process indexed by the sequence \(G_ n\), \(n\geq 1.\) It is shown that the sufficient metric entropy integral conditions cannot, in general, be substantially weakened. Using these results we easily obtain improved rates of a.s. uniform convergence for the empirical characteristic function over expanding intervals. Additionally, the rates are essentially characterized in terms of integral conditions related to the tail behavior of P.