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.