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The reverse Galton-Watson process

Published online by Cambridge University Press:  14 July 2016

Warren W. Esty
Warren W. Esty*
Affiliation:
Cleveland State University
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Abstract

Consider the following path, Zn(w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn, which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN–1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

Keywords

GALTON-WATSON PROCESSBRANCHING PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 12 , Issue 3 , September 1975 , pp. 574 - 580
Copyright
Copyright © Applied Probability Trust 1975 

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References

Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer-Verlag, Heidelberg.Google Scholar
Esty, W. W. (1973) Conditioned limit laws in critical age-dependent branching processes and some associated diffusions. Ph.D. thesis, University of Wisconsin.Google Scholar
Esty, W. W. (1975) Diffusion limits of critical branching processes conditioned on extinction in the near future. Submitted for publication.Google Scholar