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On the variance to mean ratio for random variables from Markov chains and point processes

Part of: Distribution theory - Probability Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Timothy C. Brown ,
Kais Hamza  and
Aihua Xia
Timothy C. Brown*
Affiliation:
University of Melbourne
Kais Hamza*
Affiliation:
University of Melbourne
Aihua Xia*
Affiliation:
University of New South Wales
*
∗Department of Statistics, The University of Melbourne, Parkville, VIC 3052, Australia.
∗Department of Statistics, The University of Melbourne, Parkville, VIC 3052, Australia.
∗∗∗Postal address: Department of Statistics, School of Mathematics, The University of New South Wales, Sydney 2052, Australia.
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Abstract

Criteria are determined for the variance to mean ratio to be greater than one (over-dispersed) or less than one (under-dispersed). This is done for random variables which are functions of a Markov chain in continuous time, and for the counts in a simple point process on the line. The criteria for the Markov chain are in terms of the infinitesimal generator and those for the point process in terms of the conditional intensity. Examples include a conjecture of Faddy (1994). The case of time-reversible point processes is particularly interesting, and here underdispersion is not possible. In particular, point processes which arise from Markov chains which are time-reversible, have finitely many states and are irreducible are always overdispersed.

Keywords

Markov chaintransition ratesinfinitesimal generatorpositive and negative correlationsconditional intensity

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 303 - 312
Copyright
Copyright © Applied Probability Trust 1998 

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References

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