Hermite variations of the fractional Brownian sheet

TL;DR

This paper establishes central and non-central limit theorems for Hermite variations of the anisotropic fractional Brownian sheet, depending on the Hurst parameters, revealing different asymptotic behaviors.

Contribution

It provides the first comprehensive analysis of Hermite variations for the anisotropic fractional Brownian sheet, including conditions for Gaussian and non-Gaussian limits.

Findings

Central limit theorem for certain Hurst parameters

Non-central limit theorem with Hermite process limit

Identification of parameter regimes for different limit behaviors

Abstract

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 $(1,1)$.