Example where unique stationary law, which is an occupation law, but no limit law exists I am currently learning about the balance equations, mass equation, limit law, occupation law and stationary law in Markov models. The following example is presented:

Example 2:
$$\mathcal{P} = \begin{bmatrix} 0 & 1 & 0 \\ 0.1 & 0 & 0.9 \\ 0 & 1 & 0 \end{bmatrix}.$$
The balance equations are
$$\pi_1 = 0.1 \pi_2, \ \pi_2 = \pi_1 + \pi_3, \ \pi_3 = 0.9\pi_2$$
The mass equation is
$$\pi_1 + \pi_2 + \pi_3 = 1$$
The unique solution to this balance plus mass system is
$$\pi_1 = 0.05, \ \ \ \pi_2 = 0.5, \ \ \ \pi_3 = 0.45.$$
So once again: if there is a limit law, then this is it.
However, a calculation shows that
$$\mathcal{P}^2 = \begin{bmatrix} 0.1 & 0 & 0.9 \\ 0 & 1 & 0 \\ 0.1 & 0 & 0.9 \end{bmatrix},$$
and that $\mathcal{P}^3 = \mathcal{P}$.
It follows that $\mathcal{P}^{2m - 1} = \mathcal{P}$ and $\mathcal{P}^{2m} = \mathcal{P}^2$ for $m = 1, 2, \dots$.
Thus the powers of $\mathcal{P}$ oscillate and do not converge; there is no limit law.

The following is then said:

In example 2, counting the oscillating terms shows that
$$m_{ij}(n) = \begin{cases} \delta_{ij} + \dfrac{1}{2} n(p_{ij} + p^{(2)}_{ij}) & \text{if} \ n \ \text{is even,} \\ \delta_{ij} + \dfrac{1}{2} (n + 1)p_{ij} + \dfrac{1}{2}(n - 1) p^{(2)}_{ij} & \text{if} \ n \ \text{is odd.} \end{cases}$$
Dividing by $n$, you will see that the limit exists and
$$\pi^*_{ij} = \dfrac{1}{2} (p_{ij} + p^{(2)}_{ij}) = \dfrac{1}{2}(0.1, 1, 0.9)$$
So we have a unique stationary law which is an occupation law, but no limit law exists.

There are two points of this that I am unclear on:

*

*How did the author get

$$m_{ij}(n) = \begin{cases} \delta_{ij} + \dfrac{1}{2} n(p_{ij} + p^{(2)}_{ij}) & \text{if} \ n \ \text{is even,} \\ \delta_{ij} + \dfrac{1}{2} (n + 1)p_{ij} + \dfrac{1}{2}(n - 1) p^{(2)}_{ij} & \text{if} \ n \ \text{is odd.} \end{cases}$$
from example 2?


*My understanding is that $\pi^*_{ij} = \dfrac{1}{2} (p_{ij} + p^{(2)}_{ij}) = \dfrac{1}{2}(0.1, 1, 0.9)$ is for the first (even) case (after division by $n$ and then taking the limit), but what happened to the second (odd) case?

I would greatly appreciate it if people would please take the time to clarify these two points.

EDIT:

Let $(X_n)$ be a Markov chain, and fix a state $j \in S$.
Define indicator variables: For $n = 0, 1, \dots$, let
$$I_n(j) = \begin{cases} 1 & \text{if} \ X_n = j, \\ 0 & \text{if} \ X_n \not= j. \end{cases}$$
$I_n(j) = 1$ says that the MC occupies state $j$ at time $n$.
The probability $I_n(j) = 1$ is $p^{(n)}_{ij}$ if $X_0 = i$.
$I_n (j)$ has a Bernoulli law with parameter $p^{(n)}_{ij}$.
Lemma 2. $E(I_n (j) \vert X_0 = i) = p^{(n)}_{ij}$.
Let $N_n (j) = \sum_{m = 0}^n I_m (j), \tag{6}$
$N_n (j)$ is called the occupation time of the state $j$ (up to time $n$).
Note that $\sum_{j \in S} N_n (j) = n + 1$.
The mean occupation time of state $j$, given the initial state $i$, is
$$m_{ij}(n) = E(N_n(j) \vert X_0 = i), \ \text{for all} \ i, j \in S.$$
Then $M(n) = (m_{ij}(n))_{ij}$ is called the mean occupation time matrix.
Theorem 3. The mean occupation time matrix is given by
$$M(n) = \sum_{m = 0}^n \mathcal{P}^m \tag{7}$$
Proof: It follows from Lemma 2 and (6) that
$$m_{ij}(n) = \sum_{m = 0}^n E[I_m (j) \vert X_0 = i] = \sum_{m = 0}^n p^{(m)}_{ij}.$$

$\mathcal{P}^n$ is the $n$-step transition matrix.
 A: Your second question is easier. In the limit, $\frac{n\pm 1}{n} \rightarrow 1$, so $$\frac{\delta_{ij}}{n} + \dfrac{1}{2} \frac{(n + 1)p_{ij}}{n} + \dfrac{1}{2}\frac{(n - 1) p^{(2)}_{ij}}{n}$$  goes to $\frac{1}{2}p_{ij} + \frac{1}{2}p^{(2)}_{ij}$.
For the first question: every time you go through an odd-numbered state, you increment by $p_{ij}$. This is because of theorem 3 combined with $P^n = P$ for odd $n$. Every time you go through an even-numbered state, you increment by $p^2_{ij}$. This is because of theorem 3 combined with $P^n = P^2$ for even $n$. If you write out the first few cases, you will see that's all these formulas do. For example, for $n = 5$, it's 
$$\delta_{ij} + 3p_{ij} + 2p^{(2)}_{ij}.$$ The $3$ accounts for $n = 1, 3, 5$ and the $2$ for $n = 2, 4$. The delta accounts for $n = 0$.
The intuitive explanation here is that, in terms of little cars traversing the state space, there is a continuous morning and evening rush hour between the suburbs (states $1$ and $3$) and the city (state $2$). Thus, there is no equilibrium reached at any time. But when you average over many time points, the pothole distribution does converge.
