Asymptotic properties for linear processes of functionals of reversible Markov Chains

TL;DR

This paper investigates the asymptotic behavior of linear processes driven by functionals of stationary reversible Markov chains, including long-range dependence, establishing central limit theorems and convergence to fractional Brownian motion.

Contribution

It extends asymptotic analysis to linear processes with reversible Markov chain innovations under minimal assumptions, covering long-range dependence cases.

Findings

Proves central limit theorem for the processes.

Shows convergence to fractional Brownian motion.

Handles long-range dependence in the analysis.

Abstract

In this paper we study the asymptotic behavior of linear processes having as innovations mean zero, square integrable functions of stationary reversible Markov chains. In doing so we shall preserve the generality of coefficients assuming only that they are square summable. In this way we include in our study the long range dependence case. The only assumption imposed on the innovations is the absolute summability of their covariances. Besides the central limit theorem we also study the convergence to fractional Brownian motion. The proofs are based on general results for linear processes with stationary innovations that have interest in themselves.