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Enhanced Asymptotic Analysis of Continuous-Time Markov Branching Systems: Revisiting Limiting Structural Theorems

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

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.

Enhanced Asymptotic Analysis of Continuous-Time Markov Branching Systems: Revisiting Limiting Structural Theorems

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.

Paper Structure

This paper contains 17 sections, 14 theorems, 130 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Description of the model
  4. Background and Motivation
  5. Basic Axioms and aim
  6. Basic Lemma of critical MBS theory and its consequences
  7. Convergence Theorems and Invariant Measures
  8. Conclusion
  9. Appendixes
  10. Empirical Illustration of Survival Probability in Critical MBS.
  11. On Extensions to Non-Homogeneous Systems.
  12. Enhanced Accessibility.
  13. Summary of Key Asymptotic Expansions
  14. Interpretation of the Parameter $\gamma=\delta-\nu$
  15. Special Case: $\nu=1$
  16. ...and 2 more sections

Key Result

Lemma 1

Let conditions $[{f_{\nu}}]$, $[{{\mathcal{L}}_{\nu}}]$ and $[{{\omega}_{\nu}}]$ be satisfied. Then as $t\to\infty$, where $\tau(t)={{(\nu{t})}^{1/\nu}}/{\mathcal{N}(t)}$ and $\mathcal{N}(t)$ is defined in (IIB3.2). By incorporating the supplementary constraint (IIB3.3), we directly derive the following fundamental identity: where $\rho(t;s)=o(\ln{t})$ as $t\to\infty$ uniformly in $s\in[0,1)$. $

Figures (2)

  • Figure 1: Illustrations of the asymptotic behaviour of the survival probability $q(t)$ for different slowly varying factors $\mathcal{N}(t)$ and parameter sets $(\nu,a_0)$.
  • Figure 2: Illustrations of the asymptotic behaviour of the local probability $p_1(t)$ for different slowly varying factors $\mathcal{N}(t)$ and parameter sets $(\nu,a_0)$.

Theorems & Definitions (28)

  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • Theorem 3
  • Remark 2
  • ...and 18 more