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On the properties of processes associated with a markov branching process

Published online by Cambridge University Press:  14 July 2016

V. G. Gadag  and
R. P. Gupta
V. G. Gadag*
Affiliation:
Memorial University of Newfoundland
R. P. Gupta*
Affiliation:
Dalhousie University
*
∗ Postal address: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, Canada, A1C 5S7.
∗∗ Postal address: Department of Mathematics, Statistics, and Computing, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5.
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Abstract

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ, the probability that the original process becomes extinct before τ units of time, and f(j)(qτ), the jth derivative of the offspring probability generating function f(s) at qτ when qτ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.

Keywords

MARKOV BRANCHING PROCESSSUPERCRITICALLINE OF DESCENTSPLIT-TIME PROCESSLAW OF LARGE NUMBERSSTRONG CONSISTENCY
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 520 - 528
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

On study leave from University of Poona, Pune, India.

References

Athreya, K. B. and Keiding, N. (1977) Estimation theory for continuous-time branching process. Sankhya A 39, 101–123.Google Scholar
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Gadag, V. G. and Rajarshi, M. B. (1989) On processes associated with a super-critical Markov branching process. Technical Report No. 1989–4, Statistics Division, Dalhousie University, Halifax, Canada.Google Scholar
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