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.
Categories: math.PR