The Annals of Probability

Composition semigroups and random stability

John Bunge

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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

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

First available in Project Euclid: 9 October 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60E10: Characteristic functions; other transforms

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


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

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