  The aim of this paper is to study some models of quasi-birth-and-death (QBD)
processes arising from the theory of bivariate orthogonal polynomials. First we
will see how to perform the spectral analysis in the general setting as well as
to obtain results about recurrence and the invariant measure of these processes
in terms of the spectral measure supported on some domain
$\Omega\subset\mathbb{R}^d$. Afterwards, we will apply our results to several
examples of bivariate orthogonal polynomials, namely product orthogonal
polynomials, orthogonal polynomials on a parabolic domain and orthogonal
polynomials on the triangle. We will focus on linear combinations of the Jacobi
matrices generated by these polynomials and produce families of either
continuous or discrete-time QBD processes. Finally, we show some urn models
associated with these QBD processes.
