On application of slowly varying functions with remainder in the theory of Markov branching processes with mean one and infinite variance

TL;DR

This paper explores the use of slowly varying functions in the analysis of critical Markov branching processes with infinite variance, improving foundational lemmas and refining limit theorems.

Contribution

It introduces an enhanced Basic Lemma and refined limit results for critical Markov branching processes with infinite second moments using Karamata's slowly varying functions.

Findings

Improved Basic Lemma for critical Markov branching processes

Refined limit theorems for processes with infinite variance

Application of slowly varying functions with remainder in process analysis

Abstract

We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching process. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the Basic Lemma of the theory of critical Markov branching process and refine known limit results.