An eigenvalue decomposition for first hitting times in random walks

Abstract

Distributions for first hitting times in random walks (skip-free Markov chains on the non-negative integers) are studied. Their probability generating functions are expressed in terms of the eigenvalues of probability transition submatrices. The study of first hitting times leads to several classes of infinitely divisible distributions closely related to the Bondesson class of generalized convolutions of mixtures of exponential distributions which arises from the study of first hitting times in birth-death processes. Some properties of these classes are derived from their characterization by Pick functions. © 1983 Springer-Verlag.