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July 1996 Composition semigroups and random stability
John Bunge
Ann. Probab. 24(3): 1476-1489 (July 1996). DOI: 10.1214/aop/1065725189


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.


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John Bunge. "Composition semigroups and random stability." Ann. Probab. 24 (3) 1476 - 1489, July 1996.


Published: July 1996
First available in Project Euclid: 9 October 2003

zbMATH: 0881.60013
MathSciNet: MR1411502
Digital Object Identifier: 10.1214/aop/1065725189

Primary: 60E07
Secondary: 60E10

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

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 3 • July 1996
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