Randomized Algorithms (Spring 2010)/Markov chains and random walks

Markov Chains

Stationary distributions

Theorem (Basic limit theorem) 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 [math]\displaystyle{ \lim_{n\rightarrow\infty}\pi_n(i)=\pi(i) }[/math] for all states [math]\displaystyle{ i }[/math].