Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T09:16:40.811Z Has data issue: false hasContentIssue false
  • English
  • Français

On the dependence structure of hitting times of multivariate processes

Published online by Cambridge University Press:  14 July 2016

Nader Ebrahimi  and
T. Ramallingam
Nader Ebrahimi
Affiliation:
Northern Illinois University
T. Ramallingam*
Affiliation:
Northern Illinois University
*
∗ Postal address for both authors: Division of Statistics, Department of Mathematical Sciences, Northern Illinois University, De Kalb, IL 60115, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A direct approach to derive dependence properties among the hitting times of bivariate processes has been initiated by Ebrahimi (1987) and explored further by Ebrahimi and Ramallingam (1988). In this paper, new results are obtained for multivariate processes, which help us to identify positive and negative dependence structures among the hitting times of the processes. Applications of our theorems to reliability of systems are given.

Keywords

PODNODMTP2MRR2BROWNIAN MOTIONBROWNIAN BRIDGENON-HOMOGENEOUS POISSON PROCESSCUMULATIVE DAMAGE SHOCK MODELHITTING TIMES
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 287 - 295
Copyright
Copyright © Applied Probability Trust 1989 

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

Ascher, H. and Feingold, H. (1984) Repairable Systems Reliability. Marcel Dekker, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing-Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1976) Theory of maintained systems. Math. Operat. Res. 1, 32–42.CrossRefGoogle Scholar
Ebrahimi, N. (1987) Bivariate processes with positive or negative dependent structures. J. Appl. Prob. 24, 115–122.Google Scholar
Ebrahimi, N. and Ghosh, M. (1981) Multivariate negative dependence. Commun. Statist. A10, 307–337.Google Scholar
Ebrahimi, N. and Ramallingam, T. (1988) On the dependence structure of hitting times of univariate processes. J. Appl. Prob. 25, 355–362.Google Scholar
Goel, P. and Ramallingam, T. (1987) Some properties of the maximum likelihood strategy for repairing a broken random sample. J. Statist. Planning Inf. 16, 237–248.CrossRefGoogle Scholar
Karlin, S. and Rinott, Y. (1980) Classes of orderings of measures and related correlation inequalities, I & II. J. Multivariate Anal. 10, 467–516.CrossRefGoogle Scholar
Parzen, E. (1962) Stochastic Processes. Holden-Day, San-Francisco.Google Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag, New York.Google Scholar
Shanthikumar, J. G. (1983) Software reliability models. Microelectron. Reliability 23, 903–943.CrossRefGoogle Scholar
Skorohod, A. V. (1956) Limit theorems for stochastic processes. Theory. Prob. Appl. 1, 261–290.CrossRefGoogle Scholar
Tong, Y. L. (1980) Probability Inequalities in Multivariate Distributions. Academic Press, New York.Google Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Math. Operat. Res. 5, 67–85.CrossRefGoogle Scholar