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.
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Dette, H., Fill, J.A., Pitman, J. et al. Wall and Siegmund Duality Relations for Birth and Death Chains with Reflecting Barrier. Journal of Theoretical Probability 10, 349–374 (1997). https://doi.org/10.1023/A:1022660400024
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DOI: https://doi.org/10.1023/A:1022660400024