  We consider the class of continuous-state branching processes with
immigration (CBI-processes), introduced by Kawazu and Watanabe (1971) and their
limit distributions as time tends to infinity. We determine the Levy-Khintchine
triplet of the limit distribution and give an explicit description in terms of
the characteristic triplets of the Levy subordinator and the spectrally
positive Levy process, which describe the immigration resp. branching mechanism
of the CBI-process. This representation allows us to describe the support of
the limit distribution and characterise its absolute continuity and asymptotic
behavior at the boundary of the support, generalizing several known results on
self-decomposable distributions.
