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In this paper we investigate stationary probability vectors of higher-ordertwo-dimensional symmetric transition probability tensors. We show that thereare two special symmetric transition probability tensors of order $m$ dimension2, which have and only have two stationary probability vectors; and any othersymmetric transition probability tensor of order $m$ dimension 2 has a uniquestationary probability vector. As a byproduct, we obtain that any symmetrictransition probability tensor of order $m$ dimension 2 has a unique positivestationary probability vector; and that any symmetric irreducible transitionprobability tensor of order $m$ dimension 2 has a unique stationary probabilityvector.

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