A Characterization of Reversible Markov Chains by a Rotational Representation

Abstract

Let $P = (p_{ij}), i, j = 1,2,\ldots, n$ be the matrix of a recurrent Markov chain with stationary vector $\nu > 0$ and let $R = (r_{ij}), i, j = 1,2,\ldots, n$ be a matrix, where $r_{ij} = v_ip_{ij}$. If $R$ is a symmetric matrix, we improve Alpern's rotational representation of $P$. By this representation we characterize the reversible Markov chains.