Counterexamples to the conjecture on stationary probability vectors of the second-order Markov chains

It was conjectured in the paper Stationary probability vectors of higher-order Markov chains (Li and Zhang, 2015 [7]) that if the set of stationary vectors of the second-order Markov chain contains k-interior points of the (k-1)-dimensional face of the simplex Omega(n) then every vector in the (k-1)-dimensional face is a stationary vector. In this paper, we provide counterexamples to this conjecture. (C) 2016 Elsevier Inc. All rights reserved.