Lévy Processes with Marked Jumps I: Limit Theorems

Consider a sequence $$({\tilde{Z}}_n,{\tilde{Z}}_n^{{m}})$$ ( Z ~ n , Z ~ n m ) of bivariate Lévy processes, such that $${\tilde{Z}}_n$$ Z ~ n is a spectrally positive Lévy process with finite variation, and $${\tilde{Z}}_n^{{m}}$$ Z ~ n m is the counting process of marks in $$\{0,1\}$$ { 0 , 1 } carried by the jumps of $${\tilde{Z}}_n$$ Z ~ n . 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 Lévy 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 $$({\tilde{Z}}_n,{\tilde{Z}}_n^{{m}})$$ ( 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 $$({\tilde{Z}}_n)$$ ( Z ~ n ) converges towards a Lévy process $$Z$$ 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 $$({\tilde{Z}}_n,{\tilde{Z}}_n^{{m}})$$ ( Z ~ n , Z ~ n m ) . Then, we prove the joint convergence in law of $${\tilde{Z}}_n$$ Z ~ n 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.