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Markov Chain - Is state 2 periodic?

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,145
Hey!! :eek:

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
Feb 13, 2012
1,704
Hey!! :eek:

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
Apr 14, 2013
4,145
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
Feb 13, 2012
1,704

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,145