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Weak convergence of positive self-similar Markov processes and overshoots of L\'{e}vy processes

By M. E. Caballero and L. Chaumont

Abstract

Published at http://dx.doi.org/10.1214/009117905000000611 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)Using Lamperti's relationship between L\'{e}vy processes and positive self-similar Markov processes (pssMp), we study the weak convergence of the law $\mathbb{P}_x$ of a pssMp starting at $x>0$, in the Skorohod space of c\`{a}dl\`{a}g paths, when $x$ tends to 0. To do so, we first give conditions which allow us to construct a c\`{a}dl\`{a}g 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 $\mathbb{P}_x$ 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 $x=0$

Topics: [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Publisher: Institute of Mathematical Statistics
Year: 2006
DOI identifier: 10.1214/009117905000000611
OAI identifier: oai:HAL:hal-00116354v1
Provided by: Hal-Diderot
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