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

The present article is devoted to a fine study of the convergence of renor-malized weighted quadratic and cubic variations of a fractional Brownian mo-tion 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 conver-gence 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