Behavior of the Hermite sheet with respect to the Hurst index
type: Item
instance of: scholarly article
title: Behavior of the Hermite sheet with respect to the Hurst index
zbMATH Open document ID: 1451.60052
author: Héctor Araya
author: Ciprian A. Tudor
published in: Stochastic Processes and their Applications
publication date: 2019-06-28T00:00:00Z
review text: Hermite processes are \(d\)-parameter fractional sheets with multidimensional Hurst index \(H = (H_1 , \ldots , H_d ) \in (1/2,1)^d\), which take values in the \(q\)-th Wiener chaos. They generalize the construction of fractional Brownian motion (fBm) and of the Rosenblatt process, which take values in the first and second Wiener chaos, respectively. In this paper, the weak convergence in \(C([0,T]^d)\) of Hermite processes is characterized from the convergence to \(1/2\) or to \(1\) of the components of the index \(H = (H_1 , \ldots , H_d )\). In particular, the convergence to a \(d\)-parameter centered Gaussian process with known covariance is shown when at least one component of \(H = (H_1 , \ldots , H_d )\) tends to \(1/2\). On the other hand, when at least one component of \(H = (H_1 , \ldots , H_d )\) tends to \(1\) and none of the remaining components converges to \(1/2\), the limit is non-Gaussian and described explicitly as a \(q\)-th Wiener chaos process. The proofs use the fourth moment theorem and non-central approximations of the Hermite sheet.
reviewed by: Nicolas Privault
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60G22
zbMATH DE Number: 7074620
zbMATH Keywords: Wiener chaos
zbMATH Keywords: Hermite process
zbMATH Keywords: Rosenblatt process
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: multiple stochastic integrals
zbMATH Keywords: cumulants
zbMATH Keywords: self-similarity
zbMATH Keywords: multiparameter stochastic processes
zbMATH Keywords: convergence
zbMATH Keywords: Hurst index
MaRDI profile type: Publication
full work available at URL: j.spa.2018.07.017
OpenAlex ID: W2886950120
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DOI: 10.1016/J.SPA.2018.07.017
description: scientific article; zbMATH DE number 7074620