### 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)

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
- 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
- date added to LUP
- 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}}, }