Berry-Ess\'een bounds and almost sure CLT for the quadratic variation of the bifractional Brownian motion

TL;DR

This paper establishes Berry-Esséen bounds and almost sure CLTs for the quadratic variation of bifractional Brownian motion using Malliavin calculus and Stein's method, extending understanding of convergence rates and almost sure behavior.

Contribution

It introduces new Berry-Esséen bounds and almost sure CLTs for the quadratic variation of bifractional Brownian motion within a specific parameter range, applying advanced probabilistic techniques.

Findings

Derived Berry-Esséen bounds for the quadratic variation.

Proved almost sure central limit theorems for the sequence.

Extended results to the case when 0<HK≤3/4.

Abstract

Let $B$ be a bifractional Brownian motion with parameters $H\in (0, 1)$ and $K\in(0,1]$. For any $n\geq1$, set $Z_n =\sum_{i=0}^{n-1}\big[n^{2HK}(B_{(i+1)/n}-B_{i/n})^2-\E((B_{i+1}-B_{i})^2)\big]$. We use the Malliavin calculus and the so-called Stein's method on Wiener chaos introduced by Nourdin and Peccati \cite{NP09} to derive, in the case when $0<HK\leq3/4$, Berry-Ess\'een-type bounds for the Kolmogorov distance between the law of the correct renormalization $V_n$ of $Z_n$ and the standard normal law. Finally, we study almost sure central limit theorems for the sequence $V_n$.