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

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