@article{delaporte_levy_2014,
Author = {Delaporte, Cécile},
Title = {Lévy processes with marked jumps {I}: {Limit}
theorems},
Journal = {Journal of Theoretical Probability},
Volume = {28},
Number = {4},
Pages = {1468--1499},
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.},
doi = {10.1007/s10959-014-0549-9},
issn = {0894-9840, 1572-9230},
language = {en},
month = mar,
shorttitle = {Lévy {Processes} with {Marked} {Jumps} {I}},
url = {http://link.springer.com/10.1007/s10959-014-0549-9},
urldate = {2014-08-31},
year = 2014
}