Approximating probabilities of first passage in a particular gaussian process

Shepp (1971) derives the distribution of waiting times of first passage for a particular Gaussian process. However, Shepp notes that for moderate to large waiting tines the expressions for the probability cannot be evaluated either numerically or by asymptotic estimation. Me present a useful approximation for the distribution and expected waiting time for the conditional and unconditional versions of this first passage problem. The probabilities play a role in bounds by Adler (1984) for the probability distribution of the supremum of a particular two-parameter Gaussian field, a detection problem (Lai, 197 3) and the study of signal shape problems in radars (Zakai Ziv, 1969).