Strong and weak convergence of population size in supercritical catalytic branching process

TL;DR

This paper analyzes the long-term behavior of particles in a catalytic branching process with multiple catalysts, establishing conditions for extinction, survival, and convergence of particle numbers in a supercritical setting.

Contribution

It provides new criteria for extinction and survival phases, and proves limit theorems for particle numbers in supercritical catalytic branching processes.

Findings

Established necessary and sufficient conditions for global and local extinction.

Proved almost sure and distributional convergence of normalized particle numbers.

Generalized previous results in the theory of catalytic branching processes.

Abstract

A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they may produce offspring only in the presence of catalysts located at fixed points. The asymptotic (in time) behavior of the total number of particles as well as the local particles numbers is investigated. The problems of finding the global extinction probability and local extinction probability are solved. Necessary and sufficient conditions are established for phase of pure global survival and strong local survival. Under wide conditions the limit theorems for the normalized total and local particles numbers in supercritical CBP are proved in the sense of almost surely convergence as well as with respect to convergence in distribution. Generalizations of a number of previous results are obtained as well. In the proofs the main role is played by recent results by the author devoted to classification of CBP and the moment analysis of the total and local particles numbers in CBP. Keywords and phrases: catalytic branching process, extinction probability, strong local survival, pure global phase, limit theorems, multi-type Bellman-Harris process.