1. HERMITE VARIATIONS OF THE FRACTIONAL BROWNIAN SHEET.
- Author
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RÉVEILLAC, ANTHONY, STAUCH, MICHAEL, and TUDOR, CIPRIAN A.
- Subjects
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HERMITE polynomials , *CALCULUS of variations , *WIENER processes , *CENTRAL limit theorem , *RENORMALIZATION (Physics) , *RANDOM variables , *STOCHASTIC convergence - Abstract
We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet Wα, β with Hurst parameter (α, β) ∈ (0, 1)2. When $0 \lt \alpha \leq 1-\frac{1}{2q}$ or $0 \lt \beta \leq 1-\frac{1}{2q}$ a central limit theorem holds for the renormalized Hermite variations of order q ≥ 2, while for $1-\frac{1}{2q} \lt \alpha, \beta \lt 1$ 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). [ABSTRACT FROM AUTHOR]
- Published
- 2012