Convergence of a second order Markov chain

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 FPFP as FP(x)≔Px2FP(x)≔Px2 on the n-1n-1 dimensional standard simplex ΔnΔn. If 1 is not an eigenvalue of ∇FP∇FP on ΔnΔn and P   is irreducible, then there exists a unique fixed point of FPFP on ΔnΔn. In particular, if every entry of P   is greater than 12n, then 1 is not an eigenvalue of ∇FP∇FP on ΔnΔn. Under the latter condition, we further show that the second order power method for finding the unique fixed point of FPFP on ΔnΔn is globally linearly convergent and the corresponding second order Markov process is globally R-linearly convergent.