Speed of stability for birth--death processes

TL;DR

This paper investigates the speed of convergence to stability for birth-death processes, extending previous work to non-ergodic cases, including explosive and killed processes, providing variational formulas and criteria for convergence rates.

Contribution

It introduces dual variational formulas for convergence rates of birth-death processes, covering non-ergodic, explosive, and killed cases, with practical approximation methods and criteria.

Findings

Derived dual variational formulas for convergence rates.

Established criteria for positivity of the convergence rate.

Provided approximation procedures with bounds typically within a factor of 2.

Abstract

This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or having general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which the criterion for the positivity of the rate and an approximating procedure of estimating the rate are deduced. As the first step of the approximation, the ratio of the resulting bounds is usually no more than 2. The criteria as well as basic estimates for more general types of stability are also presented. Even though the paper contributes mainly to the non-ergodic case, there are some improvements in the ergodic one. To illustrate the power of the results, a large number of examples are included.