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

摘要

The Karhunen-Loeve 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 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 established. Its Levy-Khintchine representation, and membership to the Thorin subclass of self-decomposable distributions are obtained as well. The existence and boundedness of its probability density then follow as a direct consequence.