On Functional CLT for Reversible Markov Chains with nonlinear growth of the Variance

TL;DR

This paper establishes a functional central limit theorem for stationary reversible Markov chains with non-linear variance growth, using maximal inequalities to ensure convergence to Brownian motion.

Contribution

It extends the functional CLT to cases with non-linear variance growth in reversible Markov chains, providing new conditions for convergence.

Findings

Conditional convergence implies functional CLT.

Maximal inequalities are effective for establishing tightness.

Results apply to general state space Markov chains.

Abstract

In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.