Concentration inequalities, large and moderate deviations for self-normalized empirical processes
Let \((X_n)_{n\geq 1}\) be a sequence of independent and identically distributed random variables in some measurable space \(X\), and let \(\mathcal{F}\) be a class of real-valued measurable functions on \(X\) which are centered and normalized, i.e. \(E(f(X_1))=0\) and \(E(f^2(X_1))=1\) for all \(f\in\mathcal{F}\). Let \(P_n=n^{-1}\sum_{k=1}^n\delta_{X_k}\) denote the empirical measure based on the observations \(X_1,\dots,X_n\), and consider the self-normalized empirical process \(W_n(f)=P_n(f)/\sqrt{P_n(f^2)}\) indexed by \(f\in\mathcal{F}\). For the supremum \({\mathcal W}_n=\sup_{f\in\mathcal{F}}W_n(f)\) of \(W_n\) an exponential concentration inequality is proven for certain classes \(\mathcal{F}\) of unbounded functions. This inequality is applied to establish a moderate deviation principle for \({\mathcal W}_n\). Moreover, results on large deviations of \({\mathcal W}_n\) are obtained for some parametric classes \(\mathcal{F}\) of unbounded functions.
scientific article; zbMATH DE number 1906095