AN ASYMPTOTIC FORMULA FOR THE DISTRIBUTION OF THE MAXIMUM OF A GAUSSIAN PROCESS WITH STATIONARY INCREMENTS

Let X(t), t 0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function o2(t) is convex and r2(t) = o(t) for t -0, then P(maxIojX(s)> u)- P(X(t)> u) for u ---0 and each t >0. SAMPLE FUNCTION MAXIMUM 1. Introduction and summary The main result of this paper is an asymptotic formula for the tail of the distribution of the maximum for a class of Gaussian processes X(t) with stationary increments. We assume that X(0) = 0 almost surely and EX(t) = 0 for t > 0; and put o2(t)= EX2(t). If o-2(t)= t, then X(t) is the standard Brownian motion, and the classical result of L6vy states that P(maxto. ,X(s)> u)= 2P(X(t) > u) for all u > 0 and t >0. If o2(t)= t2, then the process is of the trivial form X(t)= (t, where ( is a standard normal random variable, and so P(maxro,,tX(s)> u)= P(X(t)> u). These results suggest a similar question about the process with o-2(t)= t", for some 1 u)---P(X(t)>u), for u -> o, for every t > 0. The main result states that the distribution of the maximum is asymptotically equivalent to the distribution of the random variable observed at the terminal time. This signifies that the latter random variable is likely to be the largest in