First-Occurrence Time of High-Level Crossings in a Continuous Random Process

This paper deals with the statistical distribution of the first-occurrence and first-recurrence times of the crossing of a given level in a continuous random process. Approximate forms of the first-occurrence and first-recurrence time densities are found by considering the successive crossings to form a renewal process. A relatively simple exponential distribution is found to give an appropriate representation of the limiting case when the crossings of the level under consideration are statistically rare events. Numerical examples are worked out for some stationary Gaussian processes. The method is of use in evaluating survival probabilities for randomly excited mechanical systems subject to failure upon occurrence of a sufficiently high load.