Abstract
In the paper a weighted quadratic variation based on a sequence of partitions for a class of Gaussian processes is considered. Conditions on the sequence of partitions and the process are established for the quadratic variation to converge almost surely and for a central limit theorem to be true. Also applications to bifractional and sub-fractional Brownian motion and the estimation of their parameters are provided.
Highlights
The phenomenon of long range dependence is observed in fields such as hydrology, finance, chemistry, mathematics, physics and others, many statistical and stochastic models assuming independence or weak dependence of observations are inappropriate
The problem of the almost sure convergence of a quadratic variation has been solved for a wide class of processes with Gaussian increments by Baxter [1] and Gladyshev [2]
In order to enlarge the variety of models to choose from, extensions of fractional Brownian motion (fBm) have been introduced recently by Houdré and Villa [8] and Bojdecki et al [9]
Summary
The phenomenon of long range dependence is observed in fields such as hydrology, finance, chemistry, mathematics, physics and others, many statistical and stochastic models assuming independence or weak dependence of observations are inappropriate. Klein and Giné [3] used more general partitions and proved that particular functions of the mesh of the partition must be at most o(1/ log n) for the almost sure convergence to hold Using his result of convergence of the quadratic variation Gladyshev [2] constructed a strongly consistent estimator of H. In order to enlarge the variety of models to choose from, extensions of fBm have been introduced recently by Houdré and Villa [8] (bifractional Brownian motion) and Bojdecki et al [9] (sub-fractional Brownian motion) These processes share properties with fBm such as self similarity, gaussianity and others, they do not have stationary increments and possess some new features. We proceed further by showing in Theorem 3 that the central limit theorem holds for quadratic variation for particular values of parameters of the process
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