Markov Chains and Multiple Orthogonality

In this work we survey on connections of Markov chains and the theory of multiple orthogonality. Here we mainly concentrate on give a procedure to generate stochastic tetra diagonal Hessenberg matrices, coming from some specific families of multiple orthogonal, such as the ones of Jacobi–Piñeiro and Hypergeometric Lima–Loureiro. We show that associated with a positive tetra diagonal nonnegative bounded Hessenberg matrix we can construct two stochastic tetra diagonal ones. These two stochastic tridiagonal nonnegative Hessenberg matrices are shown to be, enlightened by the Poincaré theorem, limit transpose of each other.