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

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

2007-05-04

arxiv.org/abs/0705.0570v4

Annals of Probability 2008, Vol. 36, No. 6, 2159-2175

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AOP385 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/07-AOP385

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.

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