Finance, Economics, and Mathematics

Chapter 36Monotone Measures of Ergodicity for Markov Chains

By Julian Keilson and Oldrich Vasicek

Journal of Applied Mathematics and Stochastic Analysis, 11 (3) (1998), 283–288.

Abstract

The following paper, first written in 1974, was never published other than as part of an internal research series. Its lack of publication is certainly unrelated to the merits of the paper since the paper is of current importance by virtue of its relation to relaxation time. This chapter provides a systematic discussion of the approach of a finite Markov chain to ergodicity by proving the monotonicity of an important set of norms, each a measure of ergodicity whether or not time reversibility is present. The paper is of particular interest because the discussion of the relaxation time of a finite Markov chain (Keilson 1979) has only been clean for time reversible chains, a small subset of the chains of interest. This restriction is not present here. Indeed, a new relaxation time quoted quantifies the relaxation time for all finite ergodic chains (cf. the discussion of Q₁(t) below Eq. (7)). This relaxation time was developed by J. Keilson with A. Roy in his thesis (Roy 1996).

Introduction

Let be a finite homogeneous Markov chain in continuous time on the state space , which is irreducible and ...