# Markov Chain - Is state 2 periodic?

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

Given the Markov chain $\{X_n, n \geq 1\}$ and the following probability transition matrix:
$\begin{pmatrix} 0 & 1/3 & 2/3\\ 1/4 & 3/4 & 0\\ 2/5 & 0 & 3/5 \end{pmatrix}$

All states communicate, so the chain is irreducible, isn't?

Could you tell me if the state $2$ is periodic?

#### chisigma

##### Well-known member
Hey!!

Given the Markov chain $\{X_n, n \geq 1\}$ and the following probability transition matrix:
$\begin{pmatrix} 0 & 1/3 & 2/3\\ 1/4 & 3/4 & 0\\ 2/5 & 0 & 3/5 \end{pmatrix}$

All states communicate, so the chain is irreducible, isn't?

Could you tell me if the state $2$ is periodic?
Yes, the chain is irreducible!... the fact that $P_{2,2} \ne 0$ makes possible the return in the state 2 after any number of steps so that the state 2 is non periodic. In fact none of the states of the TM is periodic...

Kind regards

$\chi$ $\sigma$

#### mathmari

##### Well-known member
MHB Site Helper
Yes, the chain is irreducible!... the fact that $P_{2,2} \ne 0$ makes possible the return in the state 2 after any number of steps so that the state 2 is non periodic. In fact none of the states of the TM is periodic...

Kind regards

$\chi$ $\sigma$
Ok! And is the chain ergodic?

#### chisigma

##### Well-known member
Ok! And is the chain ergodic?
The MC has all aperiodic states so that it is ergodic...

Kind regards

$\chi$ $\sigma$

#### mathmari

##### Well-known member
MHB Site Helper
The MC has all aperiodic states so that it is ergodic...

Kind regards

$\chi$ $\sigma$