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On multitype processes based on progeny length of particles of a supercritical Galton-Watson process

Published online by Cambridge University Press:  14 July 2016

V. G. Gadag  and
M. B. Rajarshi
V. G. Gadag*
Affiliation:
University of Poona
M. B. Rajarshi*
Affiliation:
Pennsylvania State University
*
∗Postal address: Department of Statistics, University of Poona, Pune 411007, India.
∗∗Postal address: Department of Statistics, Pennsylvania State University, University Park, PA 16802, USA.
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Abstract

Based on infinite and finite line of descent of particles of a one-dimensional supercritical Galton-Watson branching process (GWBP), we construct an associated bivariate process. We show that this bivariate process is a two-dimensional, supercritical GWBP. We also show that this process retains its branching property on appropriate probability spaces, when conditioned on set of non-extinction and set of extinction. Some asymptotic and weak convergence results for this process have been established. A generalisation of these results to a multitype p-dimensional GWBP has also been carried out.

Keywords

FINITE LINE OF DESCENTINFINITE LINE OF DESCENTASSOCIATED PROCESSSUPER-CRITICALNON-POSITIVE REGULARNON-SINGULARCENTRAL LIMIT THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 14 - 24
Copyright
Copyright © Applied Probability Trust 1987 

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References

Aldous, D. J. and Eagleson, G. K. (1978) On mixing and stability of limit theorems. Ann. Prob. 6, 325–331.Google Scholar
Apostol, T. M. (1969) Mathematical Analysis. A Modern Approach to Advanced Calculus. Addison-Wesley, Reading, Mass.Google Scholar
Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Berlin.10.1007/978-3-642-65371-1Google Scholar
Csörgo, M. and Fischler, R. (1973) Some examples and results in the theory of mixing and random sum of central limit theorems. Period. Math. Hungar. 3, 41–67.Google Scholar
Dion, J. P. (1974) Estimation of mean and initial probabilities of a branching process. J. Appl. Prob. 11, 687–694.Google Scholar
Gadag, V. G. and Kunte, S. (1980) Some results on the non extinction paths of a super-critical Galton-Watson process. J. Indian. Statist. Assoc. 18, 73–80.Google Scholar
Gadag, V. G. and Rajarshi, M. B. (1985) On multi-type branching processes associated with a branching process. Technical Report No. 68, Department of Statistics, University of Poona, Pune, India.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
Jagers, P. (1975) Branching Processes with Biological Applications. Wiley, London.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Mode, C. J. (1971) Multitype Branching Processes. American Elsevier, New York.Google Scholar
Rényi, A. (1960) On the central limit theorem for the sum of random number of independent random variables. Acta Math. Acad. Sci. Hungar. 11, 97–102.Google Scholar
Seneta, E. (1973) Non-negative Matrices. An Introduction to Theory and Application. Allen and Unwin, London.Google Scholar