The Maximum of a Gaussian Process with Nonconstant Variance: A Sharp Bound for the Distribution Tail

Abstract

Let $X(t), 0 \leq t \leq 1$, be a real separable Gaussian process with mean 0 and continuous covariance function and put $\sigma^2(t) = EX^2(t)$. Under the well known conditions of Fernique and Dudley, the sample functions are continuous and there are explicit asymptotic upper bounds for the probability $P(\max_{\lbrack 0, 1 \rbrack}X(t) \geq u)$ for $u \rightarrow \infty$. Suppose that there is a point $\tau, 0 \leq \tau \leq 1$, such that $\sigma^2(t)$ has a unique maximum value at $t = \tau$ and put $\sigma = \sigma(\tau)$. The main result is a sharpening of the standard asymptotic upper bounds for $P(\max_{\lbrack 0, 1 \rbrack} X(t) \geq u)$ to take into account the existence of the unique maximum of $\sigma(t)$. Indeed, when the order of the standard bound exceeds that of the obvious lower bound $P(X(\tau) \geq u)$, the upper asymptotic bound is shown to be reducible by the factor $\int^1_0\exp(-u^2g(t)) dt$, with $g(t) = (1/\sigma)(1/\bar{\sigma}(t) - 1/\sigma)$, where $\bar{\sigma}(t)$ is an arbitrary majorant of $\sigma(t)$ satisfying certain general conditions. For a large class of processes the asymptotic order of the bound obtained in this way cannot be further reduced. The results are illustrated by applications to the ordinary Brownian motion and the Brownian bridge.