Sojourns and Extremes of Gaussian Processes

Let X(t), 0 t 1, be a real Gaussian process with mean 0 and continuous sample functions. For $u > 0$, form the process $u(X(t) - u)$. In this paper two related problems are studied. (i) Let G be a nonnegative measurable function, and put L = ∫¹₀ G(u(X(t) - u)) dt. For certain classes of processes X and functions G, we find, for u → ∞, the limiting conditional distribution of L given that it is positive. (ii) For the same class of processes X, we find the asymptotic form of P(max0,1 X(t) > u) for u → ∞. Finally, these results are extended to the process with the "moving barrier," $X(t) - f(t)$, where f is a continuous function.