Maximum and High Level Excursion of a Gaussian Process with Stationary Increments

Abstract

Let $X(t), t \geqq 0$, be a centered separable Gaussian process with stationary increments. Put $\sigma^2(t) = EX^2(t)$, and suppose that $\sigma(0) = 0$. Let $Y(t)$ be the normalized process $X(t)/\sigma(t)$, and $B$ an arbitrary bounded closed subinterval of the positive real axis. Under general conditions on $\sigma$ we find (1) an explicit asymptotic formula for $P\{\max_B Y > u\}$ for $u \rightarrow \infty$ in terms of $\sigma$ and various functions derived from it, and (2) the limiting conditional distribution of the time spent above the level $u$ (for $u \rightarrow \infty$) given that the time spent is positive. This limiting distribution is a scale mixture of the corresponding distribution previously obtained under comparable conditions in the case of the stationary Gaussian process.