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

TL;DR

This paper establishes limit theorems for a sequence of bivariate Le9vy processes with marked jumps, relevant for modeling genealogies of populations with mutations, and introduces a generalized ladder height process.

Contribution

It introduces a marked ladder height process for Le9vy processes with marked jumps and proves its convergence under certain conditions, extending classical results.

Findings

Convergence of the marked ladder height process in distribution.

Joint convergence of Z_n, its local time, and the marked ladder height process.

Two regimes identified for the convergence of marks.

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.