  In the probability theory limit distributions (or probability measures) are
often characterized by some convolution equations (factorization properties)
rather than by Fourier transforms (the characteristic functionals). In fact,
usually the later follows the first one. Equations, in question, involve the
multiplication by the positive scalars $c$ or an action of the corresponding
dilation $T_{c}$ on measures. In such a setting, it seems that there is no way
for stopping times (or in general, for the stochastic analysis) to come into
the "picture". However, if one accepts the view that the primary objective, in
the classical limit distributions theory, is to describe the limiting
distributions (or random variables) by the tools of random
integrals/functionals then one can use the stopping times. In this paper we
illustrate such a possibility in the case of selfdecomposability random
variables (i.e. L\'evy class L) with values in a real separable Banach space.
Also some applications of our approach to perpetuity laws are presented; cf.
\cite{Du1}, \cite{ED1}, \cite{GG1}. In fact, we show that all selfdecomposable
distributions are perpetuity laws.
