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

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.