On the rate of convergence to equilibrium for countable ergodic Markov chains

TL;DR

This paper establishes that countable ergodic Markov chains converge to equilibrium at a subgeometric rate of n^{-d} under certain initial conditions, linking convergence speed to spectral and analytic properties.

Contribution

It provides elementary proofs for subgeometric convergence rates in countable ergodic Markov chains and explores the relationship between convergence, generating functions, and spectral properties.

Findings

Convergence rate is n^{-d} for chains of ergodic degree d>0.

Explicit conditions for asymptotic convergence rates are provided.

An example with a renewal process illustrates the theoretical results.

Abstract

Using elementary methods, we prove that for a countable Markov chain $P$ of ergodic degree $d > 0$ the rate of convergence towards the stationary distribution is subgeometric of order $n^{-d}$, provided the initial distribution satisfies certain conditions of asymptotic decay. An example, modelling a renewal process and providing a markovian approximation scheme in dynamical system theory, is worked out in detail, illustrating the relationships between convergence behaviour, analytic properties of the generating functions associated to transition probabilities and spectral properties of the Markov operator $P$ on the Banach space $\ell_1$. Explicit conditions allowing to obtain the actual asymptotics for the rate of convergence are also discussed.