Consider MC with state space $\{0,1,2\}$ having the transition matrix $\begin{pmatrix} \frac{1}{2}&\frac{1}{2}&0\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{6}&\frac{1}{2}&\frac{1}{3} \end{pmatrix}$. The problem is: For each $k=0,1,2$ if the process is in state $k$, what is the average number of time periods the process will take to return to state $k$?
Does it mean that we have to find out the average return time from $0$ to $0(m_{00})$,$1$ to $1$ and $2$ to $2$?
I know that the euilibrium distribution is $(\frac{5}{14},\frac{6}{14},\frac{3}{14})$. So can I just say that the $m_{00} = \frac{1}{\frac{5}{14}}$?
For an ergodic (from transition matrix) Markov chain, the expected first return time $m_x$ for state $x$ satisfies $m_x=\frac{1}{w_x}$, where $w$ is the stationary distribution. Hence, $m_{00}=\frac{14}{5}$.