Passage times of Gaussian noise crossing a time-varying boundary

The first and second passage times of a stationary Gaussian process crossing a time-varying boundary were studied in a recent work. The results apply to stationary Gaussian processes having a finite expected rate of level crossings, but they are restricted to time-varying boundaries having zero slope at the origin. Here, the earlier results are extended to include general time-varying boundaries. Some exact, asymptotic probability densities of the first passage time of a stationary Gaussian process crossing a ramp or linear boundaries are developed. Some approximate results concerning the initial behaviors of the probability densities of the first passage times associated with a linear boundary are presented