Take a Markov Chain with state space $\left\{ 0, 1, \dots, 20 \right\}$. Then we have the rule that given $X_n$:
- Compute $Z = X_n + 1$ or $Z = X_n - 1$ with probability $\frac{1}{2}$ each (if the value is at 0 or 20, then it stays 0 or 20 with probability half and goes to 1 or 19 with probability half).
- If $X_n \ne 11$ then $X_{n+1} = Z$. If $X_n = 11$ then $X_{n+1} = 11$.
The questions are:
If $X_1 = 5$ what is the limiting distribution of this MC?
If $X_1 = 5$ or $X_1 = 15$ with probability $\frac{1}{2}$ each, what is the limiting distribution?
I think these are supposed to be different, but I get that it always just converges to the degenerate r.v 11. Am I doing something wrong?