Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 1 / 20H / b / Solution

Conditional on $X_0=x$, the random variable $T$ is the first return time $T_x^{+}$. The Kac's lemma mean-recurrence formula for a finite irreducible Markov chain givesAveraging over $X_0\sim \pi$ therefore yields