Abstract
In this paper, using the recent results on Stein’s method combining with Malliavin calculus and the almost sure central limit theorem for sequences of functionals of general Gaussian fields developed by Nourdin and Peccati, we derive the explicit bounds for the Kolmogorov distance in the central limit theorem and obtain the almost sure central limit theorem for the quadratic variation of the weighted fractional Brownian motion.
Highlights
Self-similar stochastic processes with long range dependence are of practical interest in various applications, including econometrics, Internet traffic and hydrology
The fractional Brownian motion is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data
The fBm is a suitable generalization of the standard Brownian motion, which exhibits long-range dependence, self-similarity and has stationary increments
Summary
Self-similar stochastic processes with long range dependence are of practical interest in various applications, including econometrics, Internet traffic and hydrology. The so-called wfBm Ba,b with parameters a > – , |b| < , |b| ≤ a + is a centered and self-similar Gaussian process with long/short-range dependence. It admits the relatively simple covariance as follows: s∧t. Aazizi et al [ ] and Liu [ ] studied the bi-fractional Brownian motion case, respectively Motivated by all these results, in the present work, we consider the explicit bounds for the Kolmogorov distance in the central limit theorem and almost sure central limit theorems (ASCLT in short) for the quadratic variation of wfBm. The above mentioned properties make wfBm a possible candidate for models which involve long-range dependence, self-similarity and non-stationarity. We will use ca,b and Ca,b to denote positive and finite constants depending on a, b only which may not be the same in each occurrence
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