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Hermite variations of the fractional Brownian sheet

Réveillac, Anthony; Stauch, Michael; Tudor, Ciprian A. (2012), Hermite variations of the fractional Brownian sheet, Stochastics and Dynamics, 12, 3. http://dx.doi.org/10.1142/S0219493711500213

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00522801/fr/
Date
2012
Journal name
Stochastics and Dynamics
Volume
12
Number
3
Publisher
World Scientific
Publication identifier
http://dx.doi.org/10.1142/S0219493711500213
Metadata
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Author(s)
Réveillac, Anthony
Stauch, Michael
Tudor, Ciprian A.
Abstract (EN)
We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet $W^{\alpha, \beta}$ with Hurst parameter $(\alpha, \beta) \in (0,1)^2$. When $0<\alpha \leq 1-\frac{1}{2q}$ or $0<\beta \leq 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order $q\geq 2$, while for $1-\frac{1}{2q}<\alpha, \beta < 1$ we prove that these variations satisfy a non-central limit theorem. In fact, they converge to a random variable which is the value of a two-parameter Hermite process at time $(1,1)$.
Subjects / Keywords
Limit theorems; Hermite variations; Multiple stochastic integrals; Malliavin calculus; Weak convergence

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