The Geometry of Markov Chain Limit Theorems

J. Bernhard

2013, v.19, Issue 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


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