Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T06:21:59.078Z Has data issue: false hasContentIssue false
  • English
  • Français

Conditional Poisson processes

Published online by Cambridge University Press:  14 July 2016

Richard F. Serfozo
Richard F. Serfozo*
Affiliation:
Syracuse University
Get access Rights & Permissions [Opens in a new window]

Abstract

A conditional Poisson process (often called a double stochastic Poisson process) is characterized as a random time transformation of a Poisson process with unit intensity. This characterization is used to exhibit the jump times and sizes of these processes, and to study their limiting behavior. A conditional Poisson process, whose intensity is a function of a Markov process, is discussed. Results similar to those presented can be obtained for any process with conditional stationary independent increments.

Keywords

POISSON PROCESSESRANDOM TIME TRANSFORMATIONSLIMIT THEOREMSRANDOM ENVIRONMENTSCONDITIONAL STATIONARY INDEPENDENT INCREMENTSMARKOV ENVIRONMENTSDOUBLE STOCHASTICPOINT PROCESSMARKOV ADDITIVE PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 2 , June 1972 , pp. 288 - 302
Copyright
Copyright © Applied Probability Trust 1972 

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

[1] Bartlett, M. S. (1963) The spectral analysis of point processes. J. R. Statist. Soc. B 25, 264–296.Google Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[3] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Massachusetts.Google Scholar
[4] Chung, K. L. (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.CrossRefGoogle Scholar
[5] Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
[6] Cox, D. R. (1955) Some statistical models connected with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
[7] Cox, D. R. and Lewis, P. A. W. (1966) Statistical Analysis of Series of Events. Methuen, London.Google Scholar
[8] Freedman, D. (1962) Poisson processes with random arrival rate. Ann. Math. Statist. 33, 924–929.Google Scholar
[9] Gaver, D. P. (1963) Random hazard in reliability problems. Technometrics 5, 211–225.Google Scholar
[10] Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.Google Scholar
[11] Kingman, J. F. C. (1964) On double stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.Google Scholar
[12] Mcfadden, J. A. (1965) The mixed Poisson process. Sankhya A 27, 83–92.Google Scholar
[13] Mecke, J. (1968) Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrscheinlichkeitsth. 11, 74–81.Google Scholar
[14] Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
[15] Pressman, E. L. (1968) Boundary problems for sums of lattice random variables defined on a finite regular Markov chain. Theor. Probability Appl. 12, 323–328.Google Scholar
[16] Pyke, R. and Schaufele, R. A. (1964) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.Google Scholar
[17] Serfozo, R. F. (1972) Processes with conditional stationary independent increments. J. Appl. Prob. 9, 303–315.Google Scholar
[18] Volkov, I. S. (1958) On he distribution of sums of random variables defined on a homogeneous Markov chain with finite number of states. Theor. Probability Appl. 3, 384–399.Google Scholar