Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence
Ruiz-Medina, M. Dolores
Taqqu, Murad S.
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Citation (published version)N.N. Leonenko, M.D. Ruiz-Medina, M.S. Taqqu. 2017. "Rosenblatt distribution subordinated to Gaussian random fields with long-range dependence." STOCHASTIC ANALYSIS AND APPLICATIONS, Volume 35, Issue 1, pp. 144 - 177 (34). https://doi.org/10.1080/07362994.2016.1230723
The Karhunen–Loève 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 Lévy–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.