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).
Mots-clés : Markov processes, quasi-compactness, operator, perturbation, ergodicity, Laplace transform, branching process, age-dependent process, Malthusian parameter
@article{PS_2019__23__607_0,
author = {Herv\'e, Lo{\"\i}c and Louhichi, Sana and P\`ene, Fran\c{c}oise},
title = {Multiplicative ergodicity of {Laplace} transforms for additive functional of {Markov} chains},
journal = {ESAIM: Probability and Statistics},
pages = {607--637},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2019003},
mrnumber = {4011572},
zbl = {1506.60068},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2019003/}
}
TY - JOUR AU - Hervé, Loïc AU - Louhichi, Sana AU - Pène, Françoise TI - Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains JO - ESAIM: Probability and Statistics PY - 2019 SP - 607 EP - 637 VL - 23 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2019003/ DO - 10.1051/ps/2019003 LA - en ID - PS_2019__23__607_0 ER -
%0 Journal Article %A Hervé, Loïc %A Louhichi, Sana %A Pène, Françoise %T Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains %J ESAIM: Probability and Statistics %D 2019 %P 607-637 %V 23 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2019003/ %R 10.1051/ps/2019003 %G en %F PS_2019__23__607_0
Hervé, Loïc; Louhichi, Sana; Pène, Françoise. Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 607-637. doi : 10.1051/ps/2019003. http://www.numdam.org/articles/10.1051/ps/2019003/
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