Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T13:41:10.072Z Has data issue: false hasContentIssue false
  • English
  • Français

On a class of multivariate counting processes

Part of: Special processes Applications

Published online by Cambridge University Press:  10 June 2016

Ji Hwan Cha  and
Massimiliano Giorgio
Ji Hwan Cha*
Affiliation:
Ewha Womans University
Massimiliano Giorgio*
Affiliation:
Second University of Naples
*
* Postal address: Department of Statistics, Ewha Womans University, Seoul, 120-750, Korea. Email address: [email protected]
** Postal address: Department of Industrial and Information Engineering, Second University of Naples, 81031, Aversa (CE), Italy. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we define and study a new class of multivariate counting processes, named `multivariate generalized Pólya process'. Initially, we define and study the bivariate generalized Pólya process and briefly discuss its reliability application. In order to derive the main properties of the process, we suggest some key properties and an important characterization of the process. Due to these properties and the characterization, the main properties of the bivariate generalized Pólya process are obtained efficiently. The marginal processes of the multivariate generalized Pólya process are shown to be the univariate generalized Pólya processes studied in Cha (2014). Given the history of a marginal process, the conditional property of the other process is also discussed. The bivariate generalized Pólya process is extended to the multivariate case. We define a new dependence concept for multivariate point processes and, based on it, we analyze the dependence structure of the multivariate generalized Pólya process.

Keywords

Multivariate generalized Pólya processmarginal processcomplete stochastic intensity functiondependence structureconditional counting process

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 62P30: Applications in engineering and industry
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 443 - 462
Copyright
Copyright © Applied Probability Trust 2016 

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

Allen, F. and Gale, D. (2000).Financial contagion.J. Political Econom. 108,1–34.CrossRefGoogle Scholar
Aven, T. and Jensen, U. (1999).Stochastic Models in Reliability.Springer,New York.CrossRefGoogle Scholar
Aven, T. and Jensen, U. (2000).A general minimal repair model.J. Appl. Prob. 37,187–197.CrossRefGoogle Scholar
Bowsher, C. G. (2007).Modelling security market events in continuous time: intensity based, multivariate point process models.J. Econometrics 141,876–912.Google Scholar
Cha, J. H. (2014).Characterization of the generalized Pólya process and its applications.Adv. Appl. Prob. 46,1148–1171.CrossRefGoogle Scholar
Cha, J. H. and Finkelstein, M. (2009).On a terminating shock process with independent wear increments.J. Appl. Prob. 46,353–362.CrossRefGoogle Scholar
Cha, J. H. and Lee, E. Y. (2011).A stochastic breakdown model for an unreliable web server system and an optimal admission control policy.J. Appl. Prob. 48,453–466.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1972).Multivariate point processes. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III,University of California Press,Berkeley, pp.401–448.Google Scholar
Cuadras, C. M. (2002).On the covariance between functions.J. Multivariate Anal. 81,19–27.Google Scholar
Daley, D. J. (1968).The correlation structure of the output process of some single server queueing systems.Ann. Math. Statist. 39,1007–1019.Google Scholar
Finkelstein, M. and Cha, J. H. (2013).Stochastic Modelling for Reliability: Shocks, Burn-In and Heterogeneous Populations.Springer,London.CrossRefGoogle Scholar
Joe, H. (1997).Multivariate Models and Dependence Concepts.Chapman & Hall,London.Google Scholar
Lehmann, E. L. (1966).Some concepts of dependence.Ann. Math. Statist. 37,1137–1153.CrossRefGoogle Scholar
Liang, X. and Wang, G. (2012).On a reduced form credit risk model with common shock and regime switching.Insurance Math. Econom. 51,567–575.Google Scholar
Partrat, C. (1994).Compound model for two dependent kinds of claim.Insurance Math. Econom. 15,219–231.Google Scholar
Ross, S. M. (1996).Stochastic Processes,2nd edn.John Wiley,New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007).Stochastic Orders.Springer,New York.Google Scholar