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

Erik A. van Doorn

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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.