Convergence of a second order Markov chain

Abstract

In this paper, we consider convergence properties of a second order Markov chain. Similar to a column stochastic matrix being associated to a Markov chain, a transition probability tensor P of order 3 and dimension n is associated to a second order Markov chain with n states. For this P, define FP as FP(x)≔Px2 on the n-1 dimensional standard simplex Δn. If 1 is not an eigenvalue of ∇FP on Δn and P is irreducible, then there exists a unique fixed point of FP on Δn. In particular, if every entry of P is greater than 12n, then 1 is not an eigenvalue of ∇FP on Δn. Under the latter condition, we further show that the second order power method for finding the unique fixed point of FP on Δn is globally linearly convergent and the corresponding second order Markov process is globally R-linearly convergent.