Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

Abstract

A spectral Favard theorem is derived for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The rich knowledge concerning the spectral and factorization properties of oscillatory matrices forms the basis of this theorem, which is formulated in terms of sequences of multiple orthogonal polynomials of types I and II, associated with a set of positive Lebesgue-Stieltjes measures. Additionally, a multiple Gauss quadrature is established, and the corresponding degrees of precision are determined. The spectral Favard theorem finds application in the context of Markov chains with transition matrices having (p + 2) diagonals, extending beyond the birth and death scenario, while still maintaining a positive stochastic bidiagonal factorization. In the finite case, the Karlin–McGregor spectral representation is provided, along with the demonstration of recurrent behavior and explicit expressions for the stationary distributions in terms of the orthogonal polynomials. Analogous results are obtained for countably infinite Markov chains. In this case, the Markov chain may not be recurrent, and its characterization is expressed in relation to the first measure. The ergodicity of the Markov chain is explored, taking into consideration the presence of a mass at 1, which corresponds to eigenvalues associated with the right and left eigenvectors.