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The age distribution of Markov processes

Published online by Cambridge University Press:  14 July 2016

Benny Levikson
Benny Levikson*
Affiliation:
Purdue University
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Abstract

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

Keywords

AGE DISTRIBUTIONRENEWAL THEORYBRANCHING PROCESSESBIRTH AND DEATH PROCESSESDIFFUSION APPROXIMATIONSWRIGHT-FISHER GENETICAL MODELS

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977, pp. 492 - 506
Copyright
Copyright © Applied Probability Trust 1977 

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Footnotes

This research was partially supported by NIH Grant GM 13827 through project 1669 of the Iowa Agriculture and Home Economics Experiment Station, Iowa.

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