  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.
