On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law

Assen Tchorbadjieff, Penka Mayster, Anthony G. Pakes

Research output: Contribution to journalArticlepeer-review

Abstract

The probability generating function (pgf) B(s) of the limiting conditional law (LCL) of a subcritical Markov branching process (Zt : t ≥ 0) (MBP) has a certain integral representation and it satisfies B(0) = 0 and B́(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 P1(Zt = 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.

Original languageEnglish
Pages (from-to)9-23
Number of pages15
JournalStochastics and Quality Control
Volume39
Issue number1
Early online date26 Mar 2024
DOIs
Publication statusPublished - 1 Jun 2024

Fingerprint

Dive into the research topics of 'On Subcritical Markov Branching Processes with a Specified Limiting Conditional Law'. Together they form a unique fingerprint.

Cite this