Problem 3 (15 Points)

Suppose $\{X_n\}_{n\ge0}$ is a HMC on the state space $S=\{s_1, s_2, s_3\}$ with transition matrix

$$P=\begin{pmatrix}\frac{2}{3} & \frac{1}{6} & \frac{1}{6} \\ \frac... ...3}{24} & \frac{1}{8} \\ \frac{1}{3} & \frac{1}{8} & \frac{13}{24} \end{pmatrix}$$

1. Compute the invariant distribution $\pi$
2. Show the chain is reversible.
3. Without computing $P^n$, provide a upper bound for

   $$\vert\mathbb{P}_1(X_n=s_2)-\pi_2\vert$$

   
   
   where subscript $1$ means the chain starts from state $s_1$.