Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

TL;DR

This paper studies the asymptotic behavior of quadratic variations of non-Gaussian multiparameter Hermite random fields, showing convergence to a Rosenblatt distribution under proper normalization, extending understanding of long-range dependence in these processes.

Contribution

It establishes the limit distribution of quadratic variations for multiparameter Hermite fields, generalizing known results for specific cases like fractional Brownian motion and Rosenblatt processes.

Findings

Quadratic variations converge to a Rosenblatt distribution.

Proper normalization is key for convergence.

Results apply to a broad class of non-Gaussian fields.

Abstract

Let $(Z^{q, H}_t)_{t \in [0, 1]^d}$ denote a $d$-parameter Hermite random field of order $q \geq 1$ and self-similarity parameter $H = (H_1, \ldots, H_d) \in (\frac{1}{2}, 1)^d$. This process is $H$-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion ($q=1$, $d=1$), fractional Brownian sheet $(q=1, d \geq 2)$, Rosenblatt process ($q=2$, $d=1$) as well as Rosenblatt sheet $(q=2, d \geq 2)$. For any $q \geq 2, d\geq 1$ and $H \in (\frac{1}{2}, 1)^d$ we show in this paper that a proper normalization of the quadratic variation of $Z^{q, H}$ converges in $L^2(\Omega)$ to a standard $d$-parameter Rosenblatt random variable with self-similarity index $H" = 1+ (2H-2)/q$.