Berry-Esseen bounds and almost sure CLT for the quadratic variation of a general Gaussian process

TL;DR

This paper establishes Berry-Esseen bounds and an almost sure CLT for the quadratic variation of a broad class of Gaussian processes, extending previous results to non-self-similar processes.

Contribution

It provides explicit bounds and almost sure CLT results for quadratic variations of general Gaussian processes, including non-stationary and non-self-similar cases.

Findings

Optimal Berry-Esseen bounds for H in (0, 2/3]

Upper bounds for H in (2/3, 3/4]

Almost sure CLT for H in (0, 3/4]

Abstract

In this paper, we consider the explicit bound for the second-order approximation of the quadratic variation of a general fractional Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\, s)=\mathbb{E}[G_t G_s]$ can be decomposed into two parts, one of which coincides with that of fractional Brownian motion and the other of which is bounded by $(ts)^{H-1}$ up to a constant factor. This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or have stationary increments. %Some examples include the subfractional Brownian motion and the bi-fractional Brownian motion. Under this assumption, we obtain the optimal Berry-Ess\'{e}en bounds when $H\in (0,\,\frac23]$ and the upper Berry-Ess\'{e}en bounds when $H\in (\frac23,\,\frac34]$. As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when $H\in (0,\,\frac34]$. The results extend that of \cite{NP 09} to the case of general Gaussian processes, unify and improve the Berry-Ess\'{e}en bounds in \cite{Tu 11}, \cite{AE 12} and \cite{KL 21} for respectively the sub-fractional Brownian motion, the bi-fractional Brownian motion and the sub-bifractional Brownian motion.