The One-Sided Barrier Problem for Gaussian Noise

Let X ( t ) be a real continuous p a r a m e t e r Gaussian process, s t a t i o n a r y and continuous in the m e a n . We shall assume t h r o u g h o u t t h a t EX(t) = 0 and shall write r ( r ) = EX(t)X(t + r ) . We f u r t h e r assume t h r o u g h o u t t h a t we are dealing with a separable, measurable version of the process. Our main concern in this paper is the probability P[T,r(r)] t h a t X(t) be nonnegative for 0 ^ / ^ T. T h i s q u a n t i t y is of interest as a m e a n s of describing the duration of the excursions t a k e n by the process f r o m its mean. F r o m P[T,r(r)], the distribution function /^[XJKT)] t h e interval between successive zeros of the process can be determined by differentiation [see (19)]. T h i s latter q u a n t i t y is of i m p o r t a n c e in a v a r i e t y of engineering applications of noise theory. Considerable effort has been directed in the p a s t toward the numerical determination of / ^ / ( r ) ] b o t h theoretically 1 ' 5 a n d empirically. 26 " 32 These researches h a v e resulted in various a p p r o x i m a t i o n s for F[X,r(r)], b u t m a n y of these are neither upper nor lower b o u n d s for F, a n d exact circumstances u n d e r which they are good approximations are not clear. Generally speaking, t h e y are good for small values of X a n d become nugatory for sufficiently large A. T h e r e a p p e a r s to be nothing rigorous in t h e 463