On the Limiting Probability Distribution of a Transition Probability Tensor

In this paper we propose and develop an iterative method to calculate a limiting probability distribution vector of a transition probability tensor P arising from a higher-order Markov chain. In the model, the computation of such limiting probability distribution vector x can be formulated as a Z-eigenvalue problem Pxm−1 = x associated with the eigenvalue 1 of P where all the entries of x are required to be non-negative and its summation must be equal to one. This is an analog of the matrix case for a limiting probability vector of a transition probability matrix arising from the first order Markov chain. We show that if P is a transition probability tensor, then solutions of this Z-eigenvalue problem exist. When P is irreducible, all the entries of solutions are positive. With some suitable conditions of P, the limiting probability distribution vector is even unique. Under the same uniqueness assumption, the linear convergence of the iterative method can be established. Numerical examples are presented to illustrate the theoretical results of the proposed model and the iterative method.