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 (Gt)t 0. The second order mixed partial derivative of the covariance function R(t,\, s)=E[Gt Gs] 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\'een bounds when H∈ (0,\,23] and the upper Berry-Ess\'een bounds when H∈ (23,\,34]. As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when H∈ (0,\,34]. The results extend that of NP 09 to the case of general Gaussian processes, unify and improve the Berry-Ess\'een bounds in Tu 11, AE 12 and KL 21 for respectively the sub-fractional Brownian motion, the bi-fractional Brownian motion and the sub-bifractional Brownian motion.
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