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

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