  In this paper, for $\alpha\in (1, 2}$ we show that the $\alpha$-stable
continuous-state branching process and the associated process conditioned never
to become extinct are positive self-similar Markov processes. Understanding the
interaction of the Lamperti transformation for continuous-state branching
processes and the Lamperti transformation for positive self-similar Markov
processes permits accessto a number of explicit results concerning the paths of
stable-continuous branching processes and its conditioned version.
