Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / M / 2 / c / iii / Solution

Within the class $\{1,2\}$, the two-step matrix has eigenvalues one and $1/9$, with stationary distribution $(1/2,1/2)$. Diagonalizing this symmetric matrix, or solving the recursion $p_{n+1}=4/9+p_n/9$ from $p_0=0$, givesConsequently the limit is $1/2$. Restriction to even times is essential because the original chain has period two.