Lévy Processes with Marked Jumps I: Limit Theorems

Cécile Delaporte ()
Additional contact information
Cécile Delaporte: UPMC Univ. Paris 6

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1468-1499

Abstract: Abstract 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.

Keywords: Lévy process; Invariance principle; Ladder height process; Local time at the supremum; Splitting tree; 60F17 (Primary); 60J55; 60G51 (Secondary) (search for similar items in EconPapers)
Date: 2015
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-014-0549-9 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0549-9

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-014-0549-9

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().