Wall and Siegmund Duality Relations for Birth and Death Chains with Reflecting Barrier

Holger Dette, James Allen Fill, Jim Pitman and William J. Studden
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Holger Dette: Ruhr-Universität Bochum
James Allen Fill: The Johns Hopkins University
Jim Pitman: University of California
William J. Studden: Purdue University

Journal of Theoretical Probability, 1997, vol. 10, issue 2, 349-374

Abstract: Abstract For a birth and death chain on the nonnegative integers with birth and death probabilities p i and q i≡ 1 –p i and reflecting barrier at 0, it is shown that the right tails of the probability of the first return from state 0 to state 0 are simple transition probabilities of a dual birth and death chain obtained by switching p iand q i. Combinatorial and analytical proofs are presented. Extensions and relations to other concepts of duality in the literature are discussed.

Keywords: Birth and death chain; random walk; duality; continued fraction; canonical moments (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1023/A:1022660400024

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