Showing the transition matrix is irreducible

Question

I have a 5 x 5 matrix here.

The author claims that the matrix is irreducible and using the lemma:

$If \rho_{xy}>0$ and $\rho_{yx}$, then $x$ and $y$ have the same period.

it can be shown that the state 0 and 1 are both aperiodic.

I understand the definition of irreducibility of a matrix but I am unable to go about trying to determine, using methods, to show that the matrix is irreducible in a Markov chain context. Further, a Markov chain is irreducible is it is possible to move from a state to another state (not necessarily different) in k steps -- that is, there exists a positive probability for this to occur. But moving from state 0 to state 0 or state 1 to state 1 does not provide a positive probability...which contradicts the author's claim.

Any help to aid in my understanding is appreciated.

Answer 1

To show that a Markov chain is irreducible, you need to show that you can get from any state to any other state, in some number of steps (not necessarily in one step).

In this case:

• You can get from any state to $2$ in one step, since the transition probabilities $P_{02}, P_{12}, \dots, P_{52}$ are all positive.
• You can get from state $2$ to any state either in one step (when it's $0$ or $1$) or in two steps (by going through $0$ first).

So to get from state $x$ to state $y$, one of the paths $x \to 2 \to y$ or $x \to 2 \to 0 \to y$ will have positive probability.

There are a lot of other ways to prove this, this one was just the easiest to describe.