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 L² toward an explicit limit which only depends on B. This result is somewhat surprising when compared with the celebrated Breuer and Major theorem.