Invariance principles for local times at the supremum of random walks and L\'evy processes

TL;DR

This paper establishes that local times at the supremum of converging sequences of Le9vy processes or random walks also converge, leading to joint convergence of ladder processes and applications to conditioned processes and meanders.

Contribution

It proves invariance principles for local times and ladder processes under convergence, extending known results to broader classes of processes.

Findings

Local times at the supremum converge uniformly on compact sets.

Joint convergence of ladder processes is established.

Weak convergence of conditioned random walks to the limiting process is shown.

Abstract

We prove that when a sequence of L\'evy processes $X^{(n)}$ or a normed sequence of random walks $S^{(n)}$ converges a.s. on the Skorokhod space toward a L\'evy 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.