Central and non-central limit theorems for weighted power variations of fractional Brownian motion

Abstract

In 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 in (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 L^2 to a limit which only depends on the fractional Brownian motion, and if H> 1-1/(2q), we show the convergence in L^2 to a stochastic integral with respect to the Hermite process of order q.

30 pages; minor changes

References in corpus (3)

• Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$
• Central limit theorems for multiple Skorohod integrals
• Milstein's type schemes for fractional SDEs

Cited by in corpus (7)

• Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$
• Central limit theorems for multiple Skorohod integrals
• Moments, cumulants and diagram formulae for non-linear functionals of random measures
• Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes
• Discretizing the fractional Levy area
• A strong convergence to the Rosenblatt process
• Hsu-Robbins and Spitzer's theorems for the variations of fractional Brownian motion