n this paper, we prove some central and non-central limit theorems for renormalized weighted power variations of order q≥2 of the fractional Brownian motion with Hurst parameter H∈(0, 1), where q is an integer. The central limit holds for 1/2q<H≤1−1/2q, the limit being a conditionally Gaussian distribution. If H<1/2q we show the convergence in L2 to a limit which only depends on the fractional Brownian motion, and if H>1−1/2q we show the convergence in L2 to a stochastic integral with respect to the Hermite process of order q.
This is the published version, also available here: http://dx.doi.org/10.1214/09-AIHP342.
Nourdin, Ivan., Nualart, David., Tudor, Ciprian A. "Central and non-central limit theorems for weighted power variations of fractional Brownian motion." Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 4 (2010), 1055-1079. http://dx.doi.org/10.1214/09-AIHP342.
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