Past exam of the options I course of the University of Oxford / 2011 / M / 2 / c / Solution

The finite irreducible aperiodic Markov chain converges to its stationary distribution, regardless of its initial law. Therefore $\mathbb{E}[X_j]\to {\pi }_{+}-{\pi }_{-}$. Taking the arithmetic means of this convergent sequence givesThis uses linearity of expectation and convergence of Cesàro means; it does not require the variables $X_j$ to be independent.