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Enhanced asymptotic analysis of continuous-time Markov branching systems: Revisiting limiting structural theorems

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  • Imomov, Azam A.
  • Iskandarov, Sarvar B.
  • Azimov, Jakhongir B.
  • Jumaqulov, Hurshidjon Q.

Abstract

Markov branching systems form a fundamental class of stochastic models that are extensively applied in biology, physics, finance, and other domains. These systems are distinguished by their continuous-time evolution and inherent branching structure, allowing transitions to multiple states from a single one. This branching mechanism plays a critical role in modeling phenomena such as population dynamics, epidemic spread, and probabilistic systems with multiple outcomes. Unlike standard Markov processes, branching systems require a simultaneous treatment of transition dynamics and branching probabilities, resulting in a more intricate mathematical framework. In this work, we investigate the asymptotic properties of transition functions in continuous-time Markov branching-immigration systems. Our focus lies in refining known limit theorems, establishing convergence rates, and deriving improved asymptotic expansions under relaxed moment conditions. The results contribute to a deeper understanding of the long-term behavior and invariant structures within these systems.

Suggested Citation

  • Imomov, Azam A. & Iskandarov, Sarvar B. & Azimov, Jakhongir B. & Jumaqulov, Hurshidjon Q., 2025. "Enhanced asymptotic analysis of continuous-time Markov branching systems: Revisiting limiting structural theorems," Chaos, Solitons & Fractals, Elsevier, vol. 200(P2).
    • Handle: RePEc:eee:chsofr:v:200:y:2025:i:p2:s0960077925010926
    DOI: 10.1016/j.chaos.2025.117079
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    References listed on IDEAS

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    1. Azam A. Imomov, 2014. "Limit Properties of Transition Functions of Continuous-Time Markov Branching Processes," International Journal of Stochastic Analysis, Hindawi, vol. 2014, pages 1-10, October.
    2. Chen, Anyue & Renshaw, Eric, 1993. "Recurrence of Markov branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 231-242, April.
    3. Junping Li & Anyue Chen & Anthony G. Pakes, 2012. "Asymptotic Properties of the Markov Branching Process with Immigration," Journal of Theoretical Probability, Springer, vol. 25(1), pages 122-143, March.
    4. Cohn, H. & Hering, H., 1983. "Inhomogeneous Markov branching processes: Supercritical case," Stochastic Processes and their Applications, Elsevier, vol. 14(1), pages 79-91, January.
    5. Pakes, A. G., 1976. "Some new limit theorems for the critical branching process allowing immigration," Stochastic Processes and their Applications, Elsevier, vol. 4(2), pages 175-185, April.
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