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Hervé L, Louhichi S, Pène F. Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains. ESAIM-PROBAB STAT 2019. [DOI: 10.1051/ps/2019003] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
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).
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Hervé L, Louhichi S, Pène F. Exponential growth of branching processes in a general context of lifetimes and birthtimes dependence. ESAIM-PROBAB STAT 2019. [DOI: 10.1051/ps/2019001] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
We study the exponential growth of branching 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. Applying the spectral study of Laplace-type operators recently made in Hervé et al. [ESAIM: PS 23 (2019) 607–637], we illustrate our results in the Markov context, for which the exponential growth property is linked to the Laplace transform of the lifetimes of the cells.
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