Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T13:03:22.823Z Has data issue: false hasContentIssue false
  • English
  • Français

Extreme values and crossings for the X2-Process and Other Functions of Multidimensional Gaussian Processes, by Reliability Applications

Published online by Cambridge University Press:  01 July 2016

Georg Lindgren
Georg Lindgren*
Affiliation:
University of Lund
Get access Rights & Permissions [Opens in a new window]

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

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, β = inf>x∉S ||x||, i.e. the smallest distance from the origin to an unsafe point.

Keywords

CONVERGENCE OF POINT PROCESSESEXTREMAL THEORYRELIABILITYCHI-SQUARED PROCESSESCROSSINGSMAXIMASAFETY INDEXSAFETY OF STRUCTURES
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 746 - 774
Copyright
Copyright © Applied Probability Trust 1980 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Belayev, Yu. K. (1968) On the number of exits across the boundary of a region by a vector stochastic process. Theor Prob. Appl. 13, 320–324.Google Scholar
Berman, S. M. (1971) Asymptotic independence of the number of high and low level crossings of stationary Gaussian processes. Ann. Math. Statist. 42, 927–947.Google Scholar
Bickel, P. and Rosenblatt, M. (1973) Two-dimensional random fields. In Multivariate analysis III, ed. Krishnaiah, P.-R. Academic Press, New York.Google Scholar
Hasofer, A. M. (1976) The upcrossings rate of a class of stochastic processes. In Studies in Probability and Statistics, ed. Williams, E. J., North-Holland, Amsterdam, 153–159.Google Scholar
Hasofer, A. M. and Lind, N. C. (1974) Exact and invariant second-moment code format. Journal of the Engineering Mechanics Division, ASCE 100, 111–121.CrossRefGoogle Scholar
Kallenberg, O. (1976) Random Measures. Akademie-Verlag, Berlin.Google Scholar
Leadbetter, M. R. (1974) Lectures on extreme value theory, Department of Mathematical Statistics, Lund.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1979) Extremal and related properties of stationary processes, Part I: Extremes of stationary sequences. University of Umeå, Statistical Research Report 1979–2.Google Scholar
Lind, N. C. (1977) Formulation of probabilistic design. Journal of the Engineering Mechanics Division, ASCE 103, 273–284.Google Scholar
Lindgren, G. (1980a) Model processes in non-linear prediction with application to detection and alarm. Ann. Prob. Google Scholar
Lindgren, G. (1980b) Point processes of exits by bivariate Gaussian processes, and extremal theory for the x 2-process and its concomitants. J. Multivariate Analysis. Google Scholar
Sharpe, K. (1978) Some properties of the crossings process generated by a stationary x 2 process. Adv. Appl. Prob. 10, 373–391.Google Scholar