The Geometry of Markov Chain Limit Theorems
2013, v.19, №1, 99-124
In this paper, we use a geometric viewpoint to prove several of the fundamental theorems on the convergence of Markov chains. In particular, we determine the long-term behavior of Markov chains geometrically both when they are irreducible and aperiodic and when they are not. By viewing the transition matrix of a Markov chain as a linear transformation from the standard simplex to itself, we simplify the traditional, matrix-based descriptions and proofs of the long-term behavior of Markov chains.
Keywords: Markov chain,convergence,convex polytope,geometry