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

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-Esséen bounds when $H\in (0,\frac{2}{3}]$ and the upper Berry-Esséen bounds when $H\in (\frac{2}{3},\frac{3}{4}]$. As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when $H\in (0,\frac{3}{4}]$. The results extend that of \cite{NP 09} to the case of general Gaussian processes, unify and improve the Berry-Esséen 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.

arXiv comment21 pages