An invariance principle for fractional Brownian sheets

TL;DR

This paper proves a central limit theorem and an invariance principle for high-dimensional fractional Brownian sheets, extending previous results for fractional Brownian motions using m-approximation and moment inequalities.

Contribution

It generalizes the invariance principle from fractional Brownian motions to high-dimensional fractional Brownian sheets with dependent innovations.

Findings

Established a CLT for partial sums of stationary linear random fields.

Proved an invariance principle for anisotropic fractional Brownian sheets.

Replaced martingale approximation with m-approximation in the proof.

Abstract

We establish a central limit theorem for partial sums of stationary linear random fields with dependent innovations, and an invariance principle for anisotropic fractional Brownian sheets. Our result is a generalization of the invariance principle for fractional Brownian motions by Dedecker et al. (2011) to high dimensions. A key ingredient of their argument, the martingale approximation, is replaced by an m-approximation argument. An important tool of our approach is a moment inequality for stationary random fields recently established by El Machkouri et al. (2011).