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

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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.