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

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.