Lévy Processes with Marked Jumps I: Limit Theorems
Résumé
Consider a sequence of bivariate Levy processes, such that is a spectrally positive Levy process with finite variation, and is the counting process of marks in carried by the jumps of . The study of these processes is justified by their interpretation as contour processes of a sequence of splitting trees (Lambert in Ann Probab 38(1):348-395, 2010) with mutations at birth. Indeed, this paper is the first part of a work (Delaporte in Levy processes with marked jumps II: application to a population model with mutations at birth) 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 , as a generalization of the classical ladder height process to our Levy processes with marked jumps. Assuming that the sequence converges towards a Levy process with infinite variation, we first prove the convergence in distribution, with two possible regimes for the marks, of the marked ladder height process of . Then, we prove the joint convergence in law of with its local time at the supremum and its marked ladder height process. The proof of this latter result is an adaptation of Chaumont and Doney (Ann Probab 38(4):1368-1389, 2010) to the finite variation case.