# Invariance principles for local times at the supremum of random walks and Lévy processes.

Abstract : We prove that when a sequence of Lévy processes $X^{(n)}$ or a normed sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward a Lévy process $X$, the sequence $L^{(n)}$ of local times at the supremum of $X^{(n)}$ converges uniformly on compact sets in probability toward the local time at the supremum of $X$. A consequence of this result is that the sequence of (quadrivariate) ladder processes (both ascending and descending) converges jointly in law towards the ladder processes of $X$. As an application, we show that in general, the sequence $S^{(n)}$ conditioned to stay positive converges weakly, jointly with its local time at the future minimum, towards the corresponding functional for the limiting process $X$. From this we deduce an invariance principle for the meander which extends known results for the case of attraction to a stable law.
Keywords :
Type de document :
Pré-publication, Document de travail
2009
Domaine :

Littérature citée [17 références]

https://hal.archives-ouvertes.fr/hal-00369278
Contributeur : Loïc Chaumont <>
Soumis le : mercredi 18 mars 2009 - 22:58:49
Dernière modification le : mercredi 19 décembre 2018 - 14:08:04
Document(s) archivé(s) le : mardi 8 juin 2010 - 20:40:08

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• HAL Id : hal-00369278, version 1
• ARXIV : 0903.3705

### Citation

Loïc Chaumont, Ron Doney. Invariance principles for local times at the supremum of random walks and Lévy processes.. 2009. 〈hal-00369278〉

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