Convergence of a transition probability tensor of a higher-order Markov chain to the stationary probability vector

In this paper, first we introduce a new tensor product for a transition probability tensor originating from a higher-order Markov chain. Subsequently, some properties of the new tensor product are explained, and its relationship with the stationary probability vector is studied. Also, similarity between results obtained by this new product and the first-order case is shown. Furthermore, we prove the convergence of a transition probability tensor to the stationary probability vector. Finally, we show how to achieve a stationary probability vector with some numerical examples and make some comparison between the proposed method and another existing method for obtaining stationary probability vectors. Copyright (c) 2016 John Wiley & Sons, Ltd.