# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### Extreme values and crossings for the chi^2-process and othe functions of multidimensional Gaussian processes, with reliability applications

(1980) In Advances in Applied Probability 12(3). p.746-774
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-)... (More)
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. (Less)
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
volume
12
issue
3
pages
746 - 774
publisher
Applied Probability Trust
ISSN
0001-8678
language
English
LU publication?
yes
id
70cf1107-d517-431b-b1d0-06422c63fa65 (old id 1273176)
alternative location
http://www.jstor.org/stable/1426430
2016-04-01 12:20:46
date last changed
2019-03-08 03:04:26
@article{70cf1107-d517-431b-b1d0-06422c63fa65,
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.},
author       = {Lindgren, Georg},
issn         = {0001-8678},
language     = {eng},
number       = {3},
pages        = {746--774},
publisher    = {Applied Probability Trust},
series       = {Advances in Applied Probability},
title        = {Extreme values and crossings for the chi^2-process and othe functions of multidimensional Gaussian processes, with reliability applications},
url          = {http://www.jstor.org/stable/1426430},
volume       = {12},
year         = {1980},
}