Markov Chains: Analytic and Monte Carlo Computations

4.3 Elements on the rate of convergence for laws

A modern field of study investigates the rates of convergence for the Kolmogorov ergodic theorem. The scope is to help develop and validate Monte Carlo methods for the approximate simulation from laws, which will be described in Section 5.2. One of the objectives of this section is to facilitate the reading of advanced books on the subject, such as Duflo, M. (1996) and Saloff-Coste, L. (1997). The latter book provides many examples of explicit refined bounds of convergence in law.

We are going to describe the functional analysis framework which is at the basis of the simplest aspects of these studies, which is an extension of the concepts in Section 1.3. If is finite, then powerful results can be obtained by classic tools of finite dimensional linear algebra, which they thus illustrate. If is infinite, extensions of these tools will be introduced, such as those described in the book of Rudin, W. (1991).

4.3.1 The Hilbert space framework

4.3.1.1 Fundamental Hilbert space, adjoints, and time reversal

Let be an irreducible positive recurrent Markov chain on with matrix , and be its invariant law. Consider the functional Hilbert space ...