  In this paper, three topics on semi-selfdecomposable distributions are
studied. The first one is to characterize semi-selfdecomposable distributions
by stochastic integrals with respect to Levy processes. This characterization
defines a mapping from an infinitely divisible distribution with finite
log-moment to a semi-selfdecomposable distribution. The second one is to
introduce and study a Langevin type equation and the corresponding
Ornstein-Uhlenbecktype process whose limiting distribution is
semi-selfdecomposable. Also, semi-stationary Ornstein-Uhlenbeck type processes
with semi-selfdecomposable distributions are constructed. The third one is to
study the iteration of the mapping above. The iterated mapping is expressed as
a single mapping with a different integrand. Also, nested subclasses of the
class of semi-selfdecomposable distributions are considered, andit is shown
that the limit of these nested subclasses is the closure of the class of
semi-stable distributions.
