On the Supremum of a Certain Gaussian Process

Abstract

Let $W(t), 0 \leq t \leq 1$, be the Wiener process tied down at $t = 0, t = 1; W(0) = W(1) = 0$. We find the distribution of $\sup_{0 \leq t \leq 1} W(t) - \int^1_0 W(t) dt$ in terms of the zeros of the Airy function and the positive stable density of exponent 2/3. This corresponds to the distribution of the supremum of a certain stationary, mean zero, periodic Gaussian process. It is also the limiting distribution of an optimal test statistic for the isotropy of a set of directions, proposed by G. S. Watson.