Sharper Bounds for Gaussian and Empirical Processes

Abstract

Under natural conditions on a class $\mathscr{F}$ of functions on a probability space, near optimal bounds are given for the probabilities $P\big(\sup_{f\in\mathscr{F}}|\sum_{i\leq n} f(X_i) - nE(f)| \geq M\sqrt n\big)$. The method is a variation of this author's method to study the tail probability of the supremum of a Gaussian process.