A central limit theorem for reversible processes with non-linear growth of variance

TL;DR

This paper extends the central limit theorem for reversible Markov processes with non-linear variance growth, providing conditions under which the normalized sums converge to a normal distribution, possibly non-standard.

Contribution

It introduces weaker variance growth conditions and establishes new criteria for the convergence of normalized sums to a (possibly non-standard) normal distribution.

Findings

Conditional distribution may not always converge to standard normal.

Sufficient conditions for convergence to a (possibly non-standard) normal distribution.

Examples illustrating divergence from classical CLT results.

Abstract

Kipnis and Varadhan showed that for an additive functional, $S_n$ say, of a reversible Markov chain the condition $E(S_n^{2})/n \to \kappa \in (0,\infty)$ implies the convergence of the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$, given the starting point, to the standard normal distribution. We revisit this question under the weaker condition, $E(S_n^{2}) = n\ell(n)$, where $\ell$ is a slowly varying function. It is shown by example that the conditional distribution of $S_n/\sqrt{E(S_n^{2}})$ need not converge to the standard normal distribution in this case; and sufficient conditions for convergence to a (possibly non-standard) normal distribution are developed.