Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T06:51:58.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Regenerative multivariate point processes

Published online by Cambridge University Press:  01 July 2016

Mark Berman
Mark Berman*
Affiliation:
Imperial College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

A class of stationary multivariate point processes is considered in which the events of one of the point processes act as regeneration points for the entire multivariate process. Some important properties of such processes are derived including the joint probability generating function for numbers of events in an interval of fixed length and the asymptotic behaviour of such processes. The general theory is then applied in three bivariate examples. Finally, some simple monotonicity results for stationary and renewal point processes (which are used in the second example) are proved in two appendices.

Keywords

POINT PROCESSMULTIVARIATE POINT PROCESSSTATIONARITYPOISSON PROCESSREGENERATIVE STOCHASTIC PROCESSRENEWAL PROCESSMARKOV RENEWAL PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 2 , June 1978 , pp. 411 - 430
Copyright
Copyright © Applied Probability Trust 1978 

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

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Bartlett, M. S. (1963) The spectral analysis of point processes. J.R. Statist. Soc. B 25, 264–296.Google Scholar
Berman, M. (1977) Some multivariate generalizations of results in univariate stationary point processes. J. Appl. Prob. 14. 748–757.CrossRefGoogle Scholar
Brown, M. (1978) A study of renewal processes with imrl and dfr interarrival times. Ann. Prob. 6. To appear.Google Scholar
Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1972) Multivariate point processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 401–448.Google Scholar
Daley, D. J. and Milne, R. K. (1975) Orderliness, intensities and Palm–Khinchin equations for multivariate point processes. J. Appl. Prob. 12, 383–389.Google Scholar
Gaver, D. P. (1963) Random hazard in reliability problems. Technometrics 5, 211–226.CrossRefGoogle Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Springer-Verlag, Berlin.Google Scholar
Lawrance, A. J. (1970) Selective interaction of a Poisson process and a renewal process: first-order stationary point results. J. Appl. Prob. 7, 359–372.CrossRefGoogle Scholar
Lawrance, A. J. (1971) Selective interaction of a Poisson process and a renewal process: the dependency structure of the intervals between responses. J. Appl. Prob. 8, 170–183.Google Scholar
Lewis, P. A. W. (1964) A branching Poisson process model for the analysis of computer failure patterns. J.R. Statist. Soc. B 26, 398–456.Google Scholar
Lewis, P. A. W. (1970) Asymptotic properties of branching renewal processes. IBM Research Report RC2878, Yorktown Heights, N.Y. Google Scholar
Neuts, M. F. (1971) A queue subject to extraneous phase changes. Adv. Appl. Prob. 3, 78–119.Google Scholar
Rohde, H. and Grandell, J. (1972) On the removal time of aerosol particles from the atmosphere by precipitation scavenging. Tellus 24, 443–454.Google Scholar
Rudemo, M. (1972) Doubly stochastic Poisson processes and process control. Adv. Appl. Prob. 4, 318–338.Google Scholar
Smith, W. L. (1954) Asymptotic renewal theorems. Proc. R. Soc. Edinburgh A 64, 9–48.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar
Ten Hoopen, M. and Reuver, H. A. (1965) Selective interaction of two independent recurrent processes. J. Appl. Prob. 2, 286–292.Google Scholar