# Additional aspects of the generalized linear-fractional branching process

Additional aspects of the generalized linear-fractional branching process We derive some additional results on the Bienyamé–Galton–Watson-branching process with $$\theta$$ θ -linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the $$\theta$$ θ process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the $$\theta$$ θ -branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the $$\theta$$ θ process, together with its powers. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Institute of Statistical Mathematics Springer Journals

# Additional aspects of the generalized linear-fractional branching process

, Volume 69 (5) – Jul 19, 2016
23 pages

Publisher
Springer Journals
Subject
Statistics; Statistics, general; Statistics for Business/Economics/Mathematical Finance/Insurance
ISSN
0020-3157
eISSN
1572-9052
D.O.I.
10.1007/s10463-016-0573-x
Publisher site
See Article on Publisher Site

### Abstract

We derive some additional results on the Bienyamé–Galton–Watson-branching process with $$\theta$$ θ -linear fractional branching mechanism, as studied by Sagitov and Lindo (Branching Processes and Their Applications. Lecture Notes in Statistics—Proceedings, 2016). This includes the explicit expression of the limit laws in both the subcritical cases and the supercritical cases with finite mean, and the long-run behavior of the population size in the critical case, limits laws in the supercritical cases with infinite mean when the $$\theta$$ θ process is either regular or explosive, and results regarding the time to absorption, an expression of the probability law of the $$\theta$$ θ -branching mechanism involving Bell polynomials, and the explicit computation of the stochastic transition matrix of the $$\theta$$ θ process, together with its powers.

### Journal

Annals of the Institute of Statistical MathematicsSpringer Journals

Published: Jul 19, 2016

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