On the Stationary Measure for Markov Branching Processes

Anthony G. Pakes ()
Additional contact information
Anthony G. Pakes: Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

Mathematics, 2025, vol. 13, issue 11, 1-27

Abstract: A previous study determined criteria ensuring that a probability distribution supported in positive integers is the limiting conditional law of a subcritical Markov branching process. It is known that there is an close connection between the limiting conditional law and the stationary measure of the transition semigroup. This paper revisits that theme of by seeking tractable criteria ensuring that a sequence on positive integers is the stationary measure of a subcritical or critical Markov branching process. These criteria are illustrated with several examples. The subcritical case motivates consideration of the Sibuya distribution, leading to the demonstration that members of a certain family of complete Bernstein functions, in fact, are Thorin–Bernstein. The critical case involves deriving a notion of the limiting law of population size given that extinction occurs at a precise future time. Examples are given, and some show an interesting relation between stationary measures and Hausdorff moment sequences.

Keywords: Markov branching process; limiting conditional law; stationary measure; infinite divisibility; Thorin–Bernstein function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/11/1802/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/11/1802/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:11:p:1802-:d:1666503

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().