Click here to flash read.
This paper investigates stochastic finite matrices and the corresponding
finite Markov chains constructed using recurrence matrices for general families
of orthogonal polynomials and multiple orthogonal polynomials. The paper
explores the spectral theory of transition matrices, utilizing both orthogonal
and multiple orthogonal polynomials. Several properties are derived, including
classes, periodicity, recurrence, stationary states, ergodicity, expected
recurrence times, time-reversed chains, and reversibility. Furthermore, the
paper uncovers factorization in terms of pure birth and pure death processes.
The case study focuses on hypergeometric orthogonal polynomials, where all the
computations can be carried out effectively. Particularly within the Askey
scheme, all descendants under Hahn (excluding Bessel), such as Hahn, Jacobi,
Meixner, Kravchuk, Laguerre, Charlier, and Hermite, present interesting
examples of recurrent reversible birth and death finite Markov chains.
Additionally, the paper considers multiple orthogonal polynomials, including
multiple Hahn, Jacobi-Pi\~neiro, Laguerre of the first kind, and Meixner of the
second kind, along with their hypergeometric representations and derives the
corresponding recurrent finite Markov chains and time-reversed chains.
No creative common's license