## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 4 (1981), 604-610.

### Circuit Processes

#### Abstract

Circuit processes of order $r$ are defined using a finite class of weighted circuits in a finite set $S$. The probability of the next value of the process is made proportional to the total weight of those circuits in the class which pass through the value in question and the last $r$ values. The process is an order $r$ Markov chain in $S$, and the stationary distribution is easily calculated. Also, it is shown that all stationary order $r$ Markov chains in a finite set can be represented as circuit processes of that order.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 4 (1981), 604-610.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994365

**Digital Object Identifier**

doi:10.1214/aop/1176994365

**Mathematical Reviews number (MathSciNet)**

MR624686

**Zentralblatt MATH identifier**

0464.60070

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Secondary: 60G10: Stationary processes 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

**Keywords**

05-60 Markov stationary distribution graph circuit

#### Citation

MacQueen, J. Circuit Processes. Ann. Probab. 9 (1981), no. 4, 604--610. doi:10.1214/aop/1176994365. https://projecteuclid.org/euclid.aop/1176994365