A random variable X is N-divisible if it can be
decomposed into a random sum of N i.i.d. components, where N is a random
variable independent of the components; X is N-stable if the
components are rescaled copies of X. These N-divisible and N-stable
random variables arise in a variety of stochastic models, including thinned
renewal processes and subordinated Lévy and stable processes. We
consider a general theory of N-divisibility and N-stability in the case
where $E(N) < \infty$, based on a representation of the probability
generating function of N in terms of its limiting Laplace. Stieltjes
transform $\mathscr{l}$ We analyze certain topological semigroups of such
p.g.f.’s in detail, and on this basis we extend existing
characterizations of N-divisible and N-stable laws in terms of
$\mathscr{l}$ . We apply the results to the aforementioned stochastic
models.