Let $S={\{1,2,3,4,5}\}$. Find out which states are: persistent, transient, null, non-null, periodic, aperiodic, ergodic and absorbing. Find closed and irreducible sets of a states. For closed sets that are irreducible find mean recurrence times and stationary distribution, when
$ P= \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 1/4 & 1/4 & 1/8 & 1/4 & 1/8 \\ 0 & 0 & 1/4 & 1/4 & 1/2 \\ 0 & 0 & 1/2 & 1/2 & 0 \\ 0 & 0 & 1/4 & 3/4 & 0 \end{bmatrix} $
Closed sets: ${\{3,4}\}, {\{3,5}\}$
Irreducible sets: ${\{3,4}\}, {\{3,5}\}$
Sets ${\{3,4}\}, {\{3,5}\}$ consists of states: persistent and non-null
Transient set: ${\{2}\}$
I'm not sure how to find out if a state is periodic or aperiodic. If $p_{ii}>0$, then the period equals $1$, and so all the states are periodic?
Please check the above, correct me if I'm wrong anywhere, and help me with the other part of the exercise. Any will be much appreciated.
$\{1\}$ and $\{3,4,5\}$ are closed sets, $2$ is transient and all recurrent states are aperiodic. [ $\{3,4\}$ is not a closed set because you can go from $3$ to $5$]. Of course, $1$ is an absorbing state.