A Simple Proof of Berry-Ess\'een Bounds for the Quadratic Variation of the Subfractional Brownian Motion

TL;DR

This paper presents a straightforward method to derive Berry-Esséen bounds for the quadratic variation of subfractional Brownian motion, leveraging covariance comparisons with fractional Brownian motion and existing results to improve convergence rates.

Contribution

The paper introduces a simple technique to establish Berry-Esséen bounds for subfractional Brownian motion's quadratic variation, improving convergence rate results.

Findings

Derived Berry-Esséen bounds for subfBm quadratic variation.

Bounded covariance of subfBm quadratic variation by that of fBm.

Enhanced convergence rate to match that of fBm.

Abstract

We give a simple technic to derive the Berry-Ess\'een bounds for the quadratic variation of the subfractional Brownian motion (subfBm). Our approach has two main ingredients: ($i$) bounding from above the covariance of quadratic variation of subfBm by the covariance of the quadratic variation of fractional Brownian motion (fBm); and ($ii$) using the existing results on fBm in \cite{BN08,NP09,N12}. As a result, we obtain simple and direct proof to derive the rate of convergence of quadratic variation of subfBm. In addition, we also improve this rate of convergence to meet the one of fractional Brownian motion in \cite{N12}.