Functional limit theorems for generalized variations of the fractional Brownian sheet

Mikko S. Pakkanen (1) , Anthony Réveillac (2)

1 CREATES - Center for Research in Econometric Analysis of Time Series
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We prove functional central and non-central limit theorems for generalized variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

Mots clés: Fractional Brownian sheet; central limit theorem; non-central limit theorem; Hermite sheet; power variation; Malliavin calculus

Domaines: Probabilités [math.PR]

Mikko S. Pakkanen, Anthony Réveillac. Functional limit theorems for generalized variations of the fractional Brownian sheet. 2014. ⟨hal-00976747⟩