Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-31T13:00:40.149Z Has data issue: false hasContentIssue false
  • English
  • Français

The inverses of thinned renewal processes

Published online by Cambridge University Press:  14 July 2016

Nikos Yannaros
Nikos Yannaros*
Affiliation:
The Royal Institute of Technology, Stockholm
*
∗ Postal address: Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden.
Get access Rights & Permissions [Opens in a new window]

Abstract

The class of inverses of a p-thinned renewal process is considered. It is shown that this class consists of renewal processes. It consists of Cox and renewal processes if and only if the given thinned process is Cox and renewal. In the non-Cox case, there exists a unique top renewal process, which by thinning generates all the possible inverses. Conditions for a renewal process to be a top process are given. Finally, a gamma renewal process is shown to be a top process when α > 1, where α is the shape parameter of the gamma distribution.

Keywords

POINT PROCESSTHINNED POINT PROCESSINVERSE THINNINGCOX PROCESSGAMMA RENEWAL PROCESS

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 25 , Issue 4 , December 1988 , pp. 822 - 828
Copyright
Copyright © Applied Probability Trust 1988 

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.)

Article purchase

Temporarily unavailable

References

Feller, W. (1971) An Introduction to Probability Theory and Its Applications , Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Karr, A. F. (1986) Point Processes and their Statistical Inference. Marcel Dekker, New York.Google Scholar
Kingman, J. F. C. (1964) On doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.Google Scholar
Kolsrud, T. (1986) Some comments on thinned renewal processes. Scan. Actuarial J. , (1986), 236–241.Google Scholar
Mecke, J. (1968) Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesses. Z. Wahrscheinlichkeitsth. 11, 74–81.Google Scholar
Svensson, Å. (1987) Some inequalities for thinned point processes. Dept, of Actuarial Maths and Mathematical Statistics, University of Stockholm, 1987.Google Scholar
Thedeen, T. (1986) The inverses of thinned point processes. Dept, of Statistics, University of Stockholm, Research report (1986:1).Google Scholar
Yannaros, N. (1985) On the thinning of renewal point processes. Dept, of Statistics, Univ. of Stockholm, Research report (1985:6).Google Scholar
Yannaros, N. (1987a) On Thinned Point Processes. Ph.D. dissertation, University of Stockholm.Google Scholar
Yannaros, N. (1987b) A characterization of Cox and renewal processes. Dept, of Statistics, Univ. of Stockholm, Research report (1987:4).Google Scholar
Yannaros, N. (1988) On Cox processes and gamma renewal processes. J. Appl. Prob. 25, 423–427.Google Scholar