Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

Caballero, M. E.; Chaumont, L.

Ann. Probab., Tome 34 (2006) no. 1, p. 1012-1034 / Harvested from Project Euclid

Résumé

Using Lamperti’s relationship between Lévy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law ℙ_x of a pssMp starting at x>0, in the Skorohod space of càdlàg paths, when x tends to 0. To do so, we first give conditions which allow us to construct a càdlàg Markov process X⁽⁰⁾, starting from 0, which stays positive and verifies the scaling property. Then we establish necessary and sufficient conditions for the laws ℙ_x to converge weakly to the law of X⁽⁰⁾ as x goes to 0. In particular, this answers a question raised by Lamperti [Z. Wahrsch. Verw. Gebiete 22 (1972) 205–225] about the Feller property for pssMp at x=0.

Publié le : 2006-05-14
Classification: Self-similar Markov process, Lévy process, Lamperti representation, overshoot, weak convergence, first passage time, 60G18, 60G51, 60B10

@article{1151418491,
     author = {Caballero, M. E. and Chaumont, L.},
     title = {Weak convergence of positive self-similar Markov processes and overshoots of L\'evy processes},
     journal = {Ann. Probab.},
     volume = {34},
     number = {1},
     year = {2006},
     pages = { 1012-1034},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1151418491}
}

Caballero, M. E.; Chaumont, L. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab., Tome 34 (2006) no. 1, pp.  1012-1034. http://gdmltest.u-ga.fr/item/1151418491/