In this work we survey on connections of Markov chains and the theory of multiple orthogonality. We give a procedure to generate stochastic tetradiagonal Hessenberg matrices, taking the Jacobi–Piñeiro and Hypergeometric Lima–Loureiro as a case study. We show that associated with a positive tetradiagonal nonnegative bounded Hessenberg matrix we can construct two stochastic tetradiagonal ones. These two stochastic tridiagonal nonnegative Hessenberg matrices are shown to be, enlightened by the Poincaré theorem, limit transpose of each other.