## The Annals of Probability

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

### Infinite Divisibility in Stochastic Processes

#### 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