[Math] Equivalence Classes of a Markov Chain with Transition Matrix

markov chainsmarkov-processprobabilitystochastic-processes

I have the following transition probability matrix for a markov chain with state space S={0,1,2,3,4,5,6}:

$\begin{bmatrix}
\frac13 & \frac13 &0 & 0 & \frac16 & 0 & \frac16\\
0 & \frac13 & 0 & \frac13 & 0 & \frac13 & 0\\
0 & 0 & \frac13 & 0 & \frac23 & 0 & 0\\
0 & \frac14 & 0 & \frac14 & 0 & \frac12 & 0 \\
0 & 0 & \frac12 & 0 & \frac12 & 0 & 0\\
0 & \frac12 & 0 & \frac12 & 0 & 0 & 0\\
\frac13 & \frac14 & \frac14 & 0 & 0 & \frac16 & 0
\end{bmatrix}$

Problem:

Find the equivalence classes (recurrent and transient states)!

My attempt:

{0} transient
{1,3} transient
{2,4} recurrent
{6} transient

I think this is wrong, there should only be 3 transient states.

Can someone check it please?

Best Answer

Here's a graph depiction of the Markov chain (including all possible transitions, but ignoring loops [transitions from a state to itself]).

enter image description here

So there are three equivalence classes (0 and 6 should be together, and 5 was missing from the original attempt). Also states in $\{1,3,5\}$ should be recurrent.

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