Let [math]\displaystyle{ X_0,X_1,\ldots, }[/math] be an irreducible, aperiodic Markov chain having a stationary distribution[math]\displaystyle{ \pi }[/math]. Let [math]\displaystyle{ X_0 }[/math] have the distribution [math]\displaystyle{ \pi_0 }[/math], an arbitrary initial distribution. Then