Rosenblatt distribution subordinated to gaussian random fields with long-range dependence

TL;DR

This paper derives a new Rosenblatt-type distribution for quadratic functionals of Gaussian random fields with long-range dependence, extending known results and analyzing its properties.

Contribution

It introduces a novel Rosenblatt distribution for Gaussian fields in higher dimensions, with detailed properties and representations.

Findings

Distribution reduces to Rosenblatt when d=1

Series representation in chi-squared variables provided

Distribution is self-decomposable with a bounded density

Abstract

The Karhunen-Lo\`eve expansion and the Fredholm determinant formula are used to derive an asymptotic Rosenblatt-type distribution of a sequence of integrals of quadratic functions of Gaussian stationary random fields on R^d displaying long-range dependence. This distribution reduces to the usual Rosenblatt distribution when d=1. Several properties of this new distribution are obtained. Specifically, its series representation in terms of independent chi-squared random variables is given, the asymptotic behavior of the eigenvalues, its L\`evy-Khintchine representation, as well as its membership to the Thorin subclass of self-decomposable distributions. The existence and boundedness of its probability density is then a direct consequence.