Multiplicative ergodicity of laplace transforms for additive functional of markov chains with application to age-dependent branching process

TL;DR

This paper investigates the exponential growth behavior of age-dependent branching processes with dependent lifetimes and offspring, using Laplace transforms of additive functionals of Markov chains, under weak moment assumptions.

Contribution

It introduces a novel approach linking multiplicative ergodicity of Laplace transforms to age-dependent branching processes with dependencies, expanding theoretical understanding.

Findings

Established conditions for exponential growth in dependent branching processes.

Applied results to Markov models demonstrating practical relevance.

Showed that weak moment assumptions suffice for key growth properties.

Abstract

We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).