Invariance principles for local times at the supremum of random walks and Lévy processes. - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2009

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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Dates and versions

hal-00369278 , version 1 (18-03-2009)

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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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