A Warming-Up Example

Abstract

We introduce here, in the simplest possible context, some of the ideas that will appear recurrently in the rest of the book. We first prove the classical central limit theorem for a Markov chain {X _j } _j≥0 on a countable state space, with an ergodic measure π and transition operator P. Under the condition that (I−P)⁻¹ V∈L ²(π), we prove that $\sum_{i=1}^{N} V(X_{i})/\sqrt{N}$ converges in law to a Gaussian variable with finite variance. Then, after reviewing a general central limit theorem for ergodic martingales, we prove that if π is reversible, (I−P)^−1/2 V∈L ²(π) is sufficient in order to obtain the central limit theorem. This is the central point of the Kipnis–Varadhan theorem. Notice that in this context this is also a necessary condition in order to have finite limit variance. We conclude the chapter examining the properties of the space of functions with finite limit variance, denoted, and some explicit examples.