On conditioned limit structure of the Markov branching process without finite second moment

TL;DR

This paper analyzes the limit behavior of a critical continuous-time Markov Branching Process with infinite second moment, establishing local limit theorems, ergodic properties, and introducing a new Markov Q-Process conditioned on non-extinction.

Contribution

It provides new limit theorems and properties for Markov branching processes with infinite second moments, including the construction of a novel Markov Q-Process.

Findings

Proved local limit theorems for the process

Investigated ergodic properties of the process

Introduced and studied properties of the Markov Q-Process

Abstract

Consider the continuous-time Markov Branching Process. In critical case we consider a situation when the generating function of intensity of transformation of particles has the infinite second moment, but its tail regularly varies in sense of Karamata. First we discuss limit properties of transition functions of the process. We prove local limit theorems and investigate ergodic properties of the process. Further we investigate limiting probability function conditioned to be never extinct. Hereupon we obtain a new stochastic population process as a continuous-time Markov chain called the Markov Q-Process. We study main properties of Markov Q-Process.