Extreme values and crossings for the X 2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

Abstract

Extreme values of non-linear functions of multivariate Gaussian processes are of considerable interest in engineering sciences dealing with the safety of structures. One then seeks thesurvivalprobabilitywhereX(t) = (X1(t), …,Xn(t)) is a stationary, multivariate Gaussianloadprocess, andSis asafe region.In general, the asymptotic survival probability for largeT-values is the most interesting quantity.By considering the point process formed by the extreme points of the vector processX(t), and proving a general Poisson convergence theorem, we obtain the asymptotic survival probability for a large class of safe regions, including those defined by the level curves of any second- (or higher-) degree polynomial in (x1, …,xn). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, β = inf>x∉S||x||, i.e. the smallest distance from the origin to an unsafe point.