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

• 成果类型: Article
• 署名作者: Caballero, M. E.; Chaumont, L.
• 署名单位: Universidad Nacional Autonoma de Mexico; Universite Paris Cite; Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000611
• 发表日期: 2006
• 页码: 1012-1034
• 关键词: entrance

摘要

Using Lamperti's relationship between Levy processes and positive selfsimilar Markov processes (pssMp), we study the weak convergence of the law P., of a pssMp starting at x > 0, in the Skorchod space of cadlag paths, when x tends to 0. To do so, we first give conditions which allow us to construct a cadlag 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.