Extreme values and crossings for theX²-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

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 the survival probability where X ( t ) = ( X 1 ( t ), …, X n ( t )) is a stationary, multivariate Gaussian load process, and S is a safe region. In general, the asymptotic survival probability for large T -values is the most interesting quantity. By considering the point process formed by the extreme points of the vector process X ( 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 ( x 1 , …, x n ). 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.