## The Annals of Probability

- Ann. Probab.
- Volume 24, Number 3 (1996), 1476-1489.

### Composition semigroups and random stability

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 24, Number 3 (1996), 1476-1489.

**Dates**

First available in Project Euclid: 9 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1065725189

**Digital Object Identifier**

doi:10.1214/aop/1065725189

**Mathematical Reviews number (MathSciNet)**

MR1411502

**Zentralblatt MATH identifier**

0881.60013

**Subjects**

Primary: 60E07: Infinitely divisible distributions; stable distributions

Secondary: 60E10: Characteristic functions; other transforms

**Keywords**

Composition semigroup stable distribution infinitely divisible distribution Laplace-Stieltjes transform probability generating function random stability thinned renewal process

#### Citation

Bunge, John. Composition semigroups and random stability. Ann. Probab. 24 (1996), no. 3, 1476--1489. doi:10.1214/aop/1065725189. https://projecteuclid.org/euclid.aop/1065725189