Some invariance principles for functionals of Lévy processes

Some invariance principles for functionals of Lévy processes

Mon, 16/11/2009
15:45 		Loic Chaumont (Université d’Angers) Stochastic Analysis Seminar Add to calendar Eagle House
Some invariance principles for functionals of Lévy processes
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.