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Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

  • Erik A. van Doorn (a1)
Abstract

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

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Corresponding author
Postal address: Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: e.a.vandoorn@utwente.nl
References
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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