L\'evy processes with marked jumps I : Limit theorems

Abstract

Consider a sequence (Z_n,Z_n^M) of bivariate L\'evy processes, such that Z_n is a spectrally positive L\'evy 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 L\'evy processes with marked jumps. Assuming that the sequence (Z_n) converges towards a L\'evy 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.