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

TL;DR

This paper refines the understanding of long-term behavior in continuous-time Markov branching systems by improving limit theorems and asymptotic expansions under less restrictive conditions.

Contribution

It introduces enhanced asymptotic analysis techniques for Markov branching-immigration systems, relaxing moment conditions and providing more precise convergence results.

Findings

Refined limit theorems for transition functions.

Established convergence rates under relaxed conditions.

Derived improved asymptotic expansions.

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.