Extreme values and crossings for the chi^2-process and othe functions of multidimensional Gaussian processes, with reliability applications
Lindgren, Georg LU
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1273176
- author
- Lindgren, Georg
LU
- organization
- publishing date
- 1980
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Advances in Applied Probability
- volume
- 12
- issue
- 3
- pages
- 746 - 774
- publisher
- Cambridge University Press
- 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
- date added to LUP
- 2016-04-01 12:20:46
- date last changed
- 2025-04-04 14:13:46
@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 = {{Cambridge University Press}},
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}},
}