A spectral Favard theorem for bounded banded lower Hessenberg matrices that
admit a positive bidiagonal factorization is found. The large knowledge on the
spectral and factorization properties of oscillatory matrices leads to this
spectral Favard theorem in terms of sequences of multiple orthogonal
polynomials of types I and II with respect to a set of positive
Lebesgue-Stieltjes~measures. Also a multiple Gauss quadrature is proven and
corresponding degrees of precision are found.
This spectral Favard theorem is applied to Markov chains with
$(p+2)$-diagonal transition matrices, i.e. beyond birth and death, that admit a
positive stochastic bidiagonal factorization. In the finite case, the
Karlin-McGregor spectral representation is given. It is shown that the Markov
chains are recurrent and explicit expressions in terms of the orthogonal
polynomials for the stationary distributions are given. Similar results are
obtained for the countable infinite Markov chain. Now the Markov chain is not
necessarily recurrent, and it is characterized in terms of the first measure.
Ergodicity of the Markov chain is discussed in terms of the existence of a mass
at $1$, which is an eigenvalue corresponding to the right and left
eigenvectors.
