Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion

Abstract

The present article is devoted to a fine study of the convergence of renormalized weighted quadratic and cubic variations of a fractional Brownian motion $B$ with Hurst index $H$. In the quadratic (resp. cubic) case, when $H<1/4$ (resp. $H<1/6$), we show by means of Malliavin calculus that the convergence holds in $L^2$ toward an explicit limit which only depends on $B$. This result is somewhat surprising when compared with the celebrated Breuer and Major theorem.

Published in at http://dx.doi.org/10.1214/07-AOP385 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Central limit theorems for sequences of multiple stochastic integrals

Cited by in corpus (7)

• Central limit theorems for multiple Skorohod integrals
• Asymptotic expansion of Skorohod integrals
• Discrete rough paths and limit theorems
• Estimation of quadratic variation for two-parameter diffusions
• Integration with respect to fractional local times with Hurst index $H$ greater than 1/2
• Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion
• Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets