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Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes. (English) Zbl 1163.60033

The authors consider a family of matrix valued orthogonal polynomials obtained by Pacharoni and Tirao in connection with spherical functions for the pair (SU\((N+1)\), U\((N))\); [see I. Pacharoni and J. A. Tirao, Constr. Approx., 25, No. 2, 177–192 (2007; Zbl 1132.33006)]. After an appropriate conjugation, they obtain a new family of matrix valued orthogonal polynomials where the corresponding block Jacobi matrix is stochastic and has special probabilistic properties. This gives a highly nontrivial example of a nonhomogeneous quasi-birth-and-death process for which one can explicitly compute its “\(n\)-step transition probability matrix” and its invariant distribution. The richness of the mathematical structures involved here allows us to give these explicit results for a several parameter family of quasi-birth-and-death processes with an arbitrary (finite) number of phases. Some of these results are plotted to show the effect that choices of the parameter values have on the invariant distribution.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis

Citations:

Zbl 1132.33006
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