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Conservative processes with stochastic rates

Published online by Cambridge University Press:  14 July 2016

Roy Saunders
Roy Saunders*
Affiliation:
State University of New York at Buffalo
*
*The author is presently at Northern Illinois University.
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Abstract

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

Keywords

CONSERVATIVE PROCESSESRANDOM RATESINDEPENDENT RANDOM VARIABLESEXCHANGEABLE RANDOM VARIABLESTRANSITION PROBABILITIESCARRIER-BORNE EPIDEMIC
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 3 , September 1975 , pp. 447 - 456
Copyright
Copyright © Applied Probability Trust 1975 

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Footnotes

This paper was presented at the Conference on Biomathematics and Biostatistics, Department of Pure and Applied Mathematics, Washington State University, Pullman, Washington, 1 and 2 May, 1974.

References

[1] Bartlett, M. S. (1949) Some evolutionary stochastic processes. J. R. Statist. Soc. B 11, 211–229.Google Scholar
[2] Becker, N. (1973) Carrier-epidemics in a community consisting of different groups. J. Appl. Prob. 10, 491–501.CrossRefGoogle Scholar
[3] Blum, J. R., Chernoff, H., Rosenblatt, M. and Teicher, H. (1958) Central limit theorems for interchangeable processes. Canad. J. Math. 10, 222–229.CrossRefGoogle Scholar
[4] Dietz, K. (1966) On the model of Weiss for the spread of epidemics by carriers. J. Appl. Prob. 3, 375–382.CrossRefGoogle Scholar
[5] Downton, F. (1967) Epidemics with carriers: a note on a paper of Dietz. J. Appl. Prob. 4, 264–268.CrossRefGoogle Scholar
[6] Feller, W. (1965) An Introduction to Probability Theory and Its Applications , Volume II. John Wiley and Sons, New York.Google Scholar
[7] Saunders, R. (1974) Some aspects of reversed conservative processes, carrier-borne epidemic models and phage models. Unpublished doctoral dissertation. State University of New York at Buffalo, Buffalo, New York.Google Scholar
[8] Weiss, G.H. (1965) On the spread of epidemics by carriers. Biometrics 21, 481–490.CrossRefGoogle ScholarPubMed