## The Annals of Probability

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

#### Abstract

Using Lamperti’s relationship between Lévy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law ℙx 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 ℙx 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.

#### Article information

Source
Ann. Probab. Volume 34, Number 3 (2006), 1012-1034.

Dates
First available in Project Euclid: 27 June 2006

Permanent link to this document
http://projecteuclid.org/euclid.aop/1151418491

Digital Object Identifier
doi:10.1214/009117905000000611

Mathematical Reviews number (MathSciNet)
MR2243877

#### Citation

Caballero, M. E.; Chaumont, L. Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes. Ann. Probab. 34 (2006), no. 3, 1012--1034. doi:10.1214/009117905000000611. http://projecteuclid.org/euclid.aop/1151418491.

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