A Simple Proof of Berry–Esséen Bounds for the Quadratic Variation of the Subfractional Brownian Motion

Soufiane Aazizi

doi:10.5802/ambp.358

Abstract

We give a simple technic to derive the Berry-Esseen 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 [1, 3, 2]. 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 [2].

Highlights

Introduction and preliminariesThe subfractional Brownian motion S = (St, t ≥ 0) with parameters H ∈ (0, 1), is defined on some probability space (Ω, F, P ) (Here, and everywhere else, we do assume that F is the sigma-field generated by S)
In [6], the proof uses Stein method and Malliavin calculus, based on the idea developed in [1, 4] for the case of fractional Brownian motion, which leads to the same rate of convergence
For the case of Hurst parameter H > 3/4, we think that it deserves an entire work, the quadratic variation will converge to a Hermite random variable to the fractional Brownian Motion [3]

Summary

Introduction

Nous donnons une technique simple pour calculer les limites Berry–Esséen pour la variation quadratique du mouvement Brownien subfractional (subfBm). The subfractional Brownian motion (subfBm in short) S = (St, t ≥ 0) with parameters H ∈ (0, 1), is defined on some probability space (Ω, F, P ) (Here, and everywhere else, we do assume that F is the sigma-field generated by S). The following result, proved in [6], states the convergence of quadratic variation of subfractional Brownian motion to a centered reduced normal variable, provides its rate of convergence.

Results

Conclusion