## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 5 (1980), 908-916.

### Markov Chains in Random Environments: The Case of Markovian Environments

#### Abstract

A formulation of a Markov chain in a random environment is given, generalizing special cases such as branching processes, queues, birth and death chains and random walks in random environments. It is assumed that the environmental process is Markovian, each environment corresponding to a particular law of evolution on a countable state space $\mathscr{X}$. It is then shown that there is a natural three way classification of states of $\mathscr{X}$. One of the three types of states is irregular in nature, and conditions are found under which no such states exist.

#### Article information

**Source**

Ann. Probab. Volume 8, Number 5 (1980), 908-916.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994620

**Digital Object Identifier**

doi:10.1214/aop/1176994620

**Mathematical Reviews number (MathSciNet)**

MR586775

**Zentralblatt MATH identifier**

0444.60053

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J05: Discrete-time Markov processes on general state spaces

Secondary: 60J99: None of the above, but in this section

**Keywords**

Markov chains in random environments Markovian environments classification of states inessential improperly essential properly essential bichain proper bichain

#### Citation

Cogburn, Robert. Markov Chains in Random Environments: The Case of Markovian Environments. Ann. Probab. 8 (1980), no. 5, 908--916. doi:10.1214/aop/1176994620. https://projecteuclid.org/euclid.aop/1176994620