periodicity of a Markov chain

Let $\{X_{n}\}$ be a stationary (http://planetmath.org/StationaryProcess) Markov chain with state space $I$ . Let $P_{ij}^{n}$ be the $n$ -step transition probability that the process goes from state $i$ at time $0$ to state $j$ at time $n$ :

P_{ij}^{n}=P(X_{n}=j\mid X_{0}=i).

Given any state $i\in I$ , define the set

N(i):=\{n\geq 1\mid P_{ii}^{n}>0\}.

It is not hard to see that if $n,m\in N(i)$ , then $n+m\in N(i)$ . The period of $i$ , denoted by $d(i)$ , is defined as

d(i):=\begin{cases}0&\text{if }N(i)=\varnothing,\\ \gcd(N(i))&\text{otherwise},\end{cases}

where $\gcd(N(i))$ is the greatest common divisor of all positive integers in $N(i)$ .

A state $i\in I$ is said to be aperiodic if $d(i)=1$ . A Markov chain is called aperiodic if every state is aperiodic.

Property. If states $i,j\in I$ communicate (http://planetmath.org/MarkovChainsClassStructure), then $d(i)=d(j)$ .

Proof.

We will employ a common inequality involving the $n$ -step transition probabilities:

P_{ij}^{m+n}\geq P_{ik}^{m}P_{kj}^{n}

for any $i,j,k\in I$ and non-negative integers $m, n$ .

Suppose first that $d(i)=0$ . Since $i\leftrightarrow j$ , $\displaystyle{P_{ij}^{n}>0}$ and $P_{ji}^{m}>0$ for some $n,m\geq 0$ . This implies that $P_{ii}^{m+n}>0$ , which forces $m+n=0$ or $m=n=0$ , and hence $j=i$ .

Next, assume $d(i)>0$ , this means that $N(i)\neq\varnothing$ . Since $i\leftrightarrow j$ , there are $r,s\geq 0$ such that $P_{ji}^{r}>0$ and $P_{ij}^{s}>0$ , and so $P_{jj}^{r+s}>0$ , showing $r+s\in N(j)$ . If we pick any $n\in N$ , we also have $\displaystyle{P_{jj}^{r+n+s}\geq P_{ji}^{r}P_{ii}^{n}P_{ij}^{s}>0}$ , or $r+s+n\in N(j)$ . But this means $d(j)$ divides both $r+s$ and $r+s+n$ , and so $d(j)$ divides their difference, which is $n$ . Since $n$ is arbitrarily picked, $d(j)\mid d(i)$ . Similarly, $d(i)\mid d(j)$ . Hence $d(i)=d(j)$ . ∎

Title	periodicity of a Markov chain
Canonical name	PeriodicityOfAMarkovChain
Date of creation	2013-03-22 16:24:28
Last modified on	2013-03-22 16:24:28
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60J10
Defines	period of a state
Defines	aperiodic state
Defines	aperiodic Markov chain