This article is a note for:
Volume 23, 2019
|Page(s)||607 - 637|
|Published online||26 September 2019|
Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains
INSA de Rennes,
2 IRMAR CNRS-UMR 6625, 35000 Rennes, France.
3 Université Européenne de Bretagne, Rennes, France.
4 Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38400 Saint Martin d’Hères, France.
5 Université de Brest and Institut Universitaire de France, UMR CNRS 6205, Laboratoire de Mathématique de Bretagne Atlantique, 6 avenue Le Gorgeu, 29238 Brest cedex, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 8 January 2019
This article is motivated by the quantitative study of the exponential growth of Markov-driven bifurcating processes [see Hervé et al., ESAIM: PS 23 (2019) 584–606]. In this respect, a key property is the multiplicative ergodicity, which deals with the asymptotic behaviour of some Laplace-type transform of nonnegative additive functional of a Markov chain. We establish a spectral version of this multiplicative ergodicity property in a general framework. Our approach is based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including linear autoregressive models. 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).
Mathematics Subject Classification: 60J05 / 60J85
Key words: Markov processes / quasi-compactness / operator / perturbation / ergodicity / Laplace transform / branching process / age-dependent process / Malthusian parameter
© EDP Sciences, SMAI 2019
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