2021/06/03 by Yong Chen, Zhen Ding, Chen, Yong +3
Economics, Econometrics and Finance · Mathematics · #60F05 #60G15 #60H07 #FOS: Mathematics #Mathematical Approximation and Integration #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2106.01851
openalex publication_date 2021/06/03 · openalex created_date 2021/06/22 · openalex updated_date 2026/07/28
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-Esséen bounds when H∈ (0, \frac23] and the upper Berry-Esséen bounds when H∈ (\frac23, \frac34]. As a by-product, we also show the almost sure central limit theorem (ASCLT) for the quadratic variation when H∈ (0, \frac34]. The results extend that of \citeNP 09 to the case of general Gaussian processes, unify and improve the Berry-Esséen bounds in \citeTu 11, \citeAE 12 and \citeKL 21 for respectively the sub-fractional Brownian motion, the bi-fractional Brownian motion and the sub-bifractional Brownian motion.