On exponential ergodicity and spectral structure for birth-death processes I

Abstract

Conditions for a birth-death process to be exponentially ergodic are established. It is shown that the exponential ergodicity is determined by the character of the boundary at infinity as classified by W. Feller. The spectral structure for the process is studied. Simple conditions are exhibited for the spectral span to be finite or infinite, and a simple norm is placed in evidence for the convergence. In Part I, basic relations are established and key tools of domination, log-concavity, and complete monotonicity. Feller's classification of boundaries and simple conditions for process classification are presented. Contents 1. 0. Introduction 2. 1. Birth-death processes; notation 3. 2. Basic relations between passage time and recurrence time densities and their moments 4. 3. Exponential convergence, domination, log-concavity, complete monotonicity and associated structure in the transform plane 5. 4. Structural properties of passage time densities and their transforms 6. 5. Structural properties of transition probabilities and their relation to exponential ergodicity 7. 6. Feller boundaries at infinity; process classification, simple conditions