What does this research mean for the field?

In subcritical Markov branching processes with immigration, when the Lévy measure of immigration coincides with the limit conditional law, the stationary distribution is a compound Poisson law with the same Lévy measure, though exceptions exist such as the Neyman-Scott immigration process. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To determine the relationship between the immigration flux Lévy measure, branching mechanisms, and limit conditional laws to obtain stationary distributions in subcritical Markov branching processes.

August 27, 2026Stochastics and Quality ControlOpen Access

Stationary Distribution and Limit Conditional Law in the Subcritical Markov Branching Processes with/without Homogeneous Immigration

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To determine the relationship between the immigration flux Lévy measure, branching mechanisms, and limit conditional laws to obtain stationary distributions in subcritical Markov branching processes.
Analyzed subcritical Markov branching processes with and without homogeneous immigration across Borel, logarithmic, and geometric limit conditional law distributions.
Evaluated the convergence of the extended Sibuya law to the logarithmic series and investigated Neyman–Scott immigration processes with linear death branching mechanisms.
Demonstrated that when the immigration Lévy measure matches the limit conditional law, the stationary distribution becomes a compound Poisson law with that same measure.
Identified the Neyman–Scott immigration process as a counterexample to this equivalence because its zero-truncated Poisson Lévy measure is log-concave.
Characterized the stationary distribution of the Neyman–Scott process under linear death branching using the modified exponential integral function.

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Implication

Theoretical analysis demonstrates stationary distribution behaviors in subcritical Markov branching processes, indicating how immigration flux measures determine limit conditional laws.

Cite This Study

Mayster et al. (2026) studied this question.

synapsesocial.com/papers/6a8fe98110c91c1e926210b2 https://doi.org/10.1515/eqc-2026-0045

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Implication

Theoretical analysis demonstrates stationary distribution behaviors in subcritical Markov branching processes, indicating how immigration flux measures determine limit conditional laws.

Stationary Distribution and Limit Conditional Law in the Subcritical Markov Branching Processes with/without Homogeneous Immigration

Key Points

Discussion

Implication

Cite This Study

Stationary Distribution and Limit Conditional Law in the Subcritical Markov Branching Processes with/without Homogeneous Immigration

Key Points

Discussion

Implication

Cite This Study

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