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.
"Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion." Ann. Probab. 36 (6) 2159 - 2175, November 2008. https://doi.org/10.1214/07-AOP385