# 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 PeriodicityOfAMarkovChain 2013-03-22 16:24:28 2013-03-22 16:24:28 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 60J10 period of a state aperiodic state aperiodic Markov chain