Non-central limit theorem of the weighted power variations of Gaussian processes

Abstract

By using the techniques of the Malliavin calculus, we investigate the asymptotic behavior of the weighted q-variations of continuous Gaussian process of the form Bt=∫0tK(t,s)dW(s), where W is the standard Brownian motion and K is a square integrable kernel. In particular, in the case of fractional Brownian motion with the Hurst parameter H, the limit can be expressed as the sum of q+1 Skorohod integrals of the Hermite process with self-similarity q(H−1)+1. This result gives the relation between the Skorohod integral and a pathwise Young integral of the Hermite process.