Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains

TL;DR

This paper investigates the multiplicative ergodicity of Laplace transforms of additive functionals in Markov chains, using operator perturbation methods, with applications to autoregressive models and bifurcating processes.

Contribution

It introduces a general operator perturbation approach to analyze multiplicative ergodicity without requiring exponential moment conditions.

Findings

Established multiplicative ergodicity under finite moment conditions

Applied results to linear autoregressive Markov chains

Provided a framework for studying bifurcating processes with dependence

Abstract

We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this work).