## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 4 (1987), 1339-1351.

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

Robert J. Adler and Gennady Samorodnitsky

#### 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 $\alpha = 2$.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 4 (1987), 1339-1351.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991980

**Digital Object Identifier**

doi:10.1214/aop/1176991980

**Mathematical Reviews number (MathSciNet)**

MR905335

**Zentralblatt MATH identifier**

0638.60059

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G15: Gaussian processes

Secondary: 60G57: Random measures 60F10: Large deviations 62G30: Order statistics; empirical distribution functions

**Keywords**

Gaussian processes isonormal process supremum metric entropy Brownian sheet empirical processes

#### Citation

Adler, Robert J.; Samorodnitsky, Gennady. Tail Behaviour for the Suprema of Gaussian Processes with Applications to Empirical Processes. Ann. Probab. 15 (1987), no. 4, 1339--1351. doi:10.1214/aop/1176991980. https://projecteuclid.org/euclid.aop/1176991980