On the invariance principle for reversible Markov chains

TL;DR

This paper establishes conditions under which the functional central limit theorem holds for additive functionals of reversible Markov chains, linking variance behavior to classical CLT conditions.

Contribution

It proves the equivalence between the functional CLT, variance regular variation, and the classical CLT for reversible Markov chains with general state spaces.

Findings

Functional CLT holds if and only if variance is regularly varying with exponent 1.

The CLT and conditional CLT are equivalent to the functional CLT in this setting.

Variance behavior characterizes the convergence properties of additive functionals.

Abstract

In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional central limit theorem is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.