The Annals of Probability

Infinite Divisibility in Stochastic Processes

H. D. Miller

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Abstract

It is shown that infinite divisibility of random variables, such as first passage times in a stochastic process, is often connected with the existence of an imbedded terminating renewal process. The idea is used to prove that for a continuous time Markov chain with two, three or four states all first passage times are infinitely divisible but for more than four states there are first passage times which are not infinitely divisible.

Article information

Source
Ann. Probab. Volume 7, Number 3 (1979), 406-417.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995042

Digital Object Identifier
doi:10.1214/aop/1176995042

Mathematical Reviews number (MathSciNet)
MR528319

Zentralblatt MATH identifier
0401.60014

JSTOR
links.jstor.org

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K05: Renewal theory

Keywords
First passage time infinite divisibility Markov chain terminating renewal process

Citation

Miller, H. D. Infinite Divisibility in Stochastic Processes. Ann. Probab. 7 (1979), no. 3, 406--417. doi:10.1214/aop/1176995042. https://projecteuclid.org/euclid.aop/1176995042


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