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On the Kolmogorov constant explicit form in the theory of discrete-time stochastic branching systems

Part of: Markov processes

Published online by Cambridge University Press:  04 January 2024

Azam A. Imomov Open the ORCID record for Azam A. Imomov [Opens in a new window]  and
Misliddin S. Murtazaev
Azam A. Imomov*
Affiliation:
Karshi State University
Misliddin S. Murtazaev*
Affiliation:
V. I. Romanovskiy Institute of Mathematics
*
*Postal address: 17, Kuchabag str., Karshi city 180100, Uzbekistan. Email: imomov_azam@mail.ru
**Postal address: 9, Universitet str., Almazar district, Tashkent 100174, Uzbekistan. Email: misliddin1991@mail.ru
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Abstract

We consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$. The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m<1$ on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$, where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions.

Keywords

Branching systemKolmogorov constanttransition probabilityMarkov chainbridled-survival probabilityinvariant distributionQ-processbasic lemmalimit theorems

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 15
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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