Stationary probability vectors of higher-order two-dimensional transition probability tensors

In this paper we investigate stationary probability vectors of higher-order two-dimensional symmetric transition probability tensors. We show that there are two special symmetric transition probability tensors of order $m$ dimension 2, which have and only have two stationary probability vectors; and any other symmetric transition probability tensor of order $m$ dimension 2 has a unique stationary probability vector. As a byproduct, we obtain that any symmetric transition probability tensor of order $m$ dimension 2 has a unique positive stationary probability vector; and that any symmetric irreducible transition probability tensor of order $m$ dimension 2 has a unique stationary probability vector.