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# 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.
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Cited literature [17 references]

https://hal.archives-ouvertes.fr/hal-00369278
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Submitted on : Wednesday, March 18, 2009 - 10:58:49 PM
Last modification on : Wednesday, October 20, 2021 - 3:18:46 AM
Long-term archiving on: : Tuesday, June 8, 2010 - 8:40:08 PM

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### Identifiers

• 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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