On the Second Dominant Eigenvalue Affecting the Power Method for Transition Probability Tensors

It is known that the second dominant eigenvalue of a matrix determines the convergence rate of the power method. Though ineffective for general eigenvalue computation, the power method has been of practical usage for computing the stationary distribution of a stochastic matrix. For a Markov chain with memory m, the transition “matrix" becomes an order-m tensor. Under suitable assumptions, the same power method has been used to compute the limiting probability distribution of a transition probability tensor. What is not clear is what affects the convergence rate of the iteration, if the method converges at all. Casting the power method as a fixed-point iteration, we examine the local behavior of the nonlinear map, identify the cause of convergence or divergence, and provide a family of counterexamples showing that even if the transition probability tensor is irreducible and aperiodic, the power iteration may fail to converge. AMS subject classifications. 15A18, 15A51, 15A69, 60J99