Hermite variations of the fractional Brownian sheet

2010/10/01 by Anthony Reveillac, Reveillac, Anthony, Michael Stauch +3
Mathematics · #60F05 #60H05 #91G70 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F05 #msc:60H05 #msc:91G70

paper · pdf · doi:10.48550/arxiv.1010.0143

arxiv created 2010/10/01 · arxiv updated 2010/10/04

Abstract

We prove central and non-central limit theorems for the Hermite variations of the anisotropic fractional Brownian sheet W^{α, β} with Hurst parameter (α, β) ∈ (0,1)². When 0<α≤ 1-(1)/(2q) or 0<β≤ 1-(1)/(2q) a central limit theorem holds for the renormalized Hermite variations of order q≥ 2, while for 1-(1)/(2q)<α, β< 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).

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