On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

Tchorbadjieff Assen (), Mayster Penka () and Pakes Anthony G. ()
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Tchorbadjieff Assen: Institute of Mathematics & Informatics, [87189]Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Bloc 8, 1113 Sofia, Bulgaria
Mayster Penka: Institute of Mathematics & Informatics, [87189]Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Bloc 8, 1113 Sofia, Bulgaria
Pakes Anthony G.: Department of Mathematics & Statistics, [550237]University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia

Stochastics and Quality Control, 2024, vol. 39, issue 1, 9-23

Abstract: The probability generating function (pgf) B ⁢ ( s ) {B(s)} of the limiting conditional law (LCL) of a subcritical Markov branching process ( Z t : t ≥ 0 ) {(Z_{t}:t\geq 0)} (MBP) has a certain integral representation and it satisfies B ⁢ ( 0 ) = 0 {B(0)=0} and B ′ ⁢ ( 0 ) > 0 {B^{\prime}(0)>0} . The general problem posed here is the inverse one: If a given pgf B satisfies these two conditions, is it related in this way to some MBP? We obtain some necessary conditions for this to be possible and illustrate the issues with simple examples and counterexamples. The particular case of the Borel law is shown to be the LCL of a family of MBPs and that the probabilities P 1 ( Z t = j ) {P_{1}(Z_{t}=j)} have simple explicit algebraic expressions. Exact conditions are found under which a shifted negative-binomial law can be a LCL. Finally, implications are explored for the offspring law arising from infinite divisibility of the correponding LCL.

Keywords: Subcritical Markov Branching Process; Conditional Limiting Law; Log-Series Law; Borel Law; Sibuya Law; Lambert and Cayley Functions; Lagrange Reversion; Infinite Divisibility (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1515/eqc-2023-0043

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