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arXiv · 0705.0570

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

Ivan Nourdin
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Abstract

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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Ivan Nourdin. 2009-01-19. Asymptotic behavior of weighted quadratic and cubic variations of fractional Brownian motion. https://doi.org/10.1214/07-aop385

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