Lévy processes with marked jumps I : Limit theorems

Cécile Delaporte 1, *
* Auteur correspondant
1 Probabilités, statistiques et biologie
LPMA - Laboratoire de Probabilités et Modèles Aléatoires
Abstract : Consider a sequence (Z_n,Z_n^M) of bivariate Lévy processes, such that Z_n is a spectrally positive Lévy process with finite variation, and Z_n^M is the counting process of marks in {0,1} carried by the jumps of Z_n. The study of these processes is justified by their interpretation as contour processes of a sequence of splitting trees with mutations at birth. Indeed, this paper is the first part of a work aiming to establish an invariance principle for the genealogies of such populations enriched with their mutational histories. To this aim, we define a bivariate subordinator that we call the marked ladder height process of (Z_n,Z_n^M), as a generalization of the classical ladder height process to our Lévy processes with marked jumps. Assuming that the sequence (Z_n) converges towards a Lévy process Z with infinite variation, we first prove the convergence in distribution, with two possible regimes for the marks, of the marked ladder height process of (Z_n,Z_n^M). Then we prove the joint convergence in law of Z_n with its local time at the supremum and its marked ladder height process.
Type de document :
Pré-publication, Document de travail
27 pages. 2013
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https://hal.archives-ouvertes.fr/hal-00828480
Contributeur : Cécile Delaporte <>
Soumis le : vendredi 31 mai 2013 - 10:38:25
Dernière modification le : lundi 29 mai 2017 - 14:21:54

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  • HAL Id : hal-00828480, version 1
  • ARXIV : 1305.6245

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Cécile Delaporte. Lévy processes with marked jumps I : Limit theorems. 27 pages. 2013. 〈hal-00828480〉

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