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 the survival probability P{(X1(t),⋯ ,Xn(t))∈ S, all t∈ [0,T]}, where X(t)=(X1(t),⋯ ,Xn(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 (x1,⋯ ,xn). This makes it possible to give an asymptotic theory for the so-called Hasofer-Lind reliability index, $\beta =\text{inf}_{x\not\in S}\|x\|$, i.e. the smallest distance from the origin to an unsafe point.
Original language | English |
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Pages (from-to) | 746-774 |
Journal | Advances in Applied Probability |
Volume | 12 |
Issue number | 3 |
Publication status | Published - 1980 |
Subject classification (UKÄ)
- Probability Theory and Statistics