**Question**

Consider the following transition matrix:

` P= 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1/4 1/4 0 1/2 0 0 1 0 0 0 0 0 0 1/3 0 0 0 2/3 `

**a)** Which states are transient?

**b)** Which states are recurrent?

**c)** Identify all closed sets of states.

**d)** Is this chain ergodic?

Dear friends ,

I have thought the states are as follows,

{1,5} {0,2,4} recurrent since they communicate with each other

{3} transient

and since state 3 does not communicate with other states, it is not an ergodic Mc

Am I correct?

Thank you..

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#### Best Answer

Let the state space of the Markov Chain be $S={1,2,3,4,5,6}$. Now draw the state transition diagram.

(a). From the figure, we observe that ${4}$, and ${6}$ form non-closed communicating classes. State $2$ does not communicate even with itself and such a state is called a **non-return** state. Hence, the states **2**, **4** and **6** are **transient**.

(b)&(c). The class ${1,3,5}$ is a closed-communicating class. Hence, states **1**, **3** and **5** are recurrent states.

(c). There is only one closed-communicating class, ${1,3,5}$.

(d). As the chain is not an irreducible Markov Chain, it is not an ergodic chain.