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Spectral analysis of bilateral birth–death processes: some new explicit examples

Part of: Markov processes Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  15 June 2022

Manuel D. de la Iglesia
Manuel D. de la Iglesia*
Affiliation:
Universidad Nacional Autónoma de México
*
*Postal address: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Ciudad de México, Mexico. Email address: mdi29@im.unam.mx
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Abstract

We consider the spectral analysis of several examples of bilateral birth–death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, such as recurrence, the invariant distribution (if it exists), and the probability current.

Keywords

Bilateral birth–death processesorthogonal polynomialsspectral analysis

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05: Orthogonal functions and polynomials, general theory
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 1193 - 1221
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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