First Outcrossing Probability Bounds

Abstract

The structural reliability theory encounters the difficult problem of assessing the probability that a random vector process crosses out of a given safe domain within a given time interval. For stationary vector processes, a way of calculating an optimal lower bound is demonstrated. The method is illustrated on examples of scalar Gaussian processes. For small outcrossing probabilities, the lower bounds are quite close to the well‐known upper bounds based on the mean outcrossing rate. The comparison with some general asymptotic results valid for stationary Gaussian processes and the comparison with the only known exact result (due to Slepian) valid for a particular Gaussian process reveal some peculiar problems of evaluating the outcrossing probability for Gaussian processes with an infinite mean outcrossing rate. These types of processes seem questionable as civil engineering models when directly used in a threshold crossing failure criterion without first being passed through a suitable smoothing filter.

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