Abstract:
Fractional iteration of probability generating functions is investigated. In particular, conditions on the generating function of a Galton–Watson process that are necessary and sufficient for the process to admit imbedding in a continuous-time homogeneous Markov branching process are obtained. Necessary imbedding conditions formulated in terms of the several initial coefficients of the generating function are also obtained. The collection of all probability generating functions is partitioned, in accordance with a classification of branching processes, into subsets, and the latter are described as convex hulls of their extreme points. A description is given of the infinitesimal generators of distinguished semigroups of probability generating functions.
Citation:
V. V. Goryainov, “Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes”, Russian Acad. Sci. Sb. Math., 79:1 (1994), 47–61
\Bibitem{Gor93}
\by V.~V.~Goryainov
\paper Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 79
\issue 1
\pages 47--61
\mathnet{http://mi.mathnet.ru/eng/sm986}
\crossref{https://doi.org/10.1070/SM1994v079n01ABEH003488}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1239751}
\zmath{https://zbmath.org/?q=an:0809.60090}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1994PP19200004}
Linking options:
https://www.mathnet.ru/eng/sm986
https://doi.org/10.1070/SM1994v079n01ABEH003488
https://www.mathnet.ru/eng/sm/v184/i5/p55
This publication is cited in the following 7 articles:
Pavel Gumenyuk, Takahiro Hasebe, José-Luis Pérez, “Loewner Theory for Bernstein Functions I: Evolution Families and Differential Equations”, Constr Approx, 2024
V. V. Goryainov, O. S. Kudryavtseva, A. P. Solodov, “Iterates of holomorphic maps, fixed points, and domains of univalence”, Russian Math. Surveys, 77:6 (2022), 959–1020
V. V. Goryainov, “Semigroups of analytic functions in analysis and applications”, Russian Math. Surveys, 67:6 (2012), 975–1021
Joffe A., Letac G., “Multitype Linear Fractional Branching Processes”, J. Appl. Probab., 43:4 (2006), 1091–1106
V. V. Goryainov, “Koenigs function and fractional iterates of probability generating functions”, Sb. Math., 193:7 (2002), 1009–1025
A. V. Shipileva, “Estimates of the Distribution of the Extinction Moment of a Markov Branching Process”, Theory Probab Appl, 45:4 (2001), 695
A. V. Shipileva, “Limit distributions for branching processes with immigration”, Russian Math. (Iz. VUZ), 44:1 (2000), 76–82
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