Recurrent extensions of self-similar Markov processes and Cramér’s condition II

Let P = (Fx, x > 0) be a family of probability measures on Skohorod's space D+, the space of cadlag paths defined on [0, oof with values in R+. The space D+ is endowed with the Skohorod topology and its Borel a-field. We will denote by X the canonical process of the coordinates and (Qt, t > 0) will be the natural filtration generated by X. Assume that under P the canonical process X is a positive self-similar Markov process (pssMp). That is, (X, P) is a [0, oo [-valued strong Markov process with the following scaling property: there exists an a > 0 such that for every c > 0,

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