Jacobi–Piñeiro Markov chains

Given a non-negative recursion matrix describing higher order recurrence relations for multiple orthogonal polynomials of type II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The Karlin–McGregor represen...

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Bibliographic Details
Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2024-01, Vol.118 (1), Article 15
Main Authors: Branquinho, Amílcar, Díaz, Juan E. F., Foulquié-Moreno, Ana, Mañas, Manuel, Álvarez-Fernández, Carlos
Format: Article
Language:English
Subjects:
Applications of Mathematics
Markov analysis
Markov chains
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Parameters
Polynomials
Theoretical
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Summary:Given a non-negative recursion matrix describing higher order recurrence relations for multiple orthogonal polynomials of type II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The Karlin–McGregor representation formula is extended to both dual Markov chains and applied to the discussion of the corresponding generating functions and first-passage distributions. Recurrent or transient character of the Markov chain is discussed. The Jacobi–Piñeiro multiple orthogonal polynomials are taken as a case study of the described results. The region of parameters where the recursion matrix is non-negative is given. Moreover, two stochastic matrices, describing two dual Markov chains are given in terms of the recursion matrix and the values of the multiple orthogonal polynomials of type II and corresponding linear forms of type I at the point  x = 1 . The region of parameters where the Markov chains are recurrent or transient is given, and the connection between both dual Markov chains is discussed at the light of the Poincaré’s theorem.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-023-01510-x