Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem

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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 Karlin and McGregor’s representation for the transition probabilities of a birth-death process, and the Courant-Fischer Theorem for eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.
Original languageUndefined
Place of PublicationEnschede
PublisherUniversity of Twente, Department of Applied Mathematics
Number of pages18
Publication statusPublished - Jan 2014

Publication series

NameMemorandum
PublisherUniversity of Twente, Department of Applied Mathematics
No.2033
ISSN (Print)1874-4850
ISSN (Electronic)1874-4850

Keywords

  • METIS-303990
  • Exponential decay
  • IR-88727
  • Birth-death process
  • Rate of convergence
  • Orthogonal polynomials
  • EWI-24292

Cite this

van Doorn, E. A. (2014). Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem. (Memorandum; No. 2033). Enschede: University of Twente, Department of Applied Mathematics.
van Doorn, Erik A. / Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem. Enschede : University of Twente, Department of Applied Mathematics, 2014. 18 p. (Memorandum; 2033).
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keywords = "METIS-303990, Exponential decay, IR-88727, Birth-death process, Rate of convergence, Orthogonal polynomials, EWI-24292",
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van Doorn, EA 2014, Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem. Memorandum, no. 2033, University of Twente, Department of Applied Mathematics, Enschede.

Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem. / van Doorn, Erik A.

Enschede : University of Twente, Department of Applied Mathematics, 2014. 18 p. (Memorandum; No. 2033).

Research output: Book/ReportReportProfessional

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T1 - Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem

AU - van Doorn, Erik A.

PY - 2014/1

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N2 - 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 Karlin and McGregor’s representation for the transition probabilities of a birth-death process, and the Courant-Fischer Theorem for eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

AB - 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 Karlin and McGregor’s representation for the transition probabilities of a birth-death process, and the Courant-Fischer Theorem for eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

KW - METIS-303990

KW - Exponential decay

KW - IR-88727

KW - Birth-death process

KW - Rate of convergence

KW - Orthogonal polynomials

KW - EWI-24292

M3 - Report

T3 - Memorandum

BT - Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem

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van Doorn EA. Representations for the decay parameter of a birth-death process based on the Courant-Fischer Theorem. Enschede: University of Twente, Department of Applied Mathematics, 2014. 18 p. (Memorandum; 2033).