Tail Behaviour for the Suprema of Gaussian Processes with Applications to Empirical Processes

Abstract

Initially we consider "the" standard isonormal linear process $L$ on a Hilbert space $H$, and applying metric entropy methods obtain bounds for the probability that $\sup_CLx > \lambda, C \subset H$ and $\lambda$ large. Under the assumption that the entropy function of $C$ grows polynomially, we find bounds of the form $c\lambda^\alpha\exp(- \frac{1}{2}\lambda^2/\sigma^2)$, where $\sigma^2$ is the maximal variance of $L$. We use a notion of entropy finer than that usually employed and specifically suited to the nonstationary situation. As a result we obtain, in the nonstationary setting, more precise bounds than any in the literature. We then treat a number of examples in which the power $\alpha$ is identified. These include the distributions of the maxima of the rectangle indexed, pinned Brownian sheet on $\mathbb{R}^k$ for which $\alpha = 2(2k - 1)$, and the half plane indexed pinned sheet on $\mathbb{R}^2$ for which $\alpha = 2$.