Non-central limit theorems for random fields subordinated to gamma-correlated random fields

TL;DR

This paper establishes non-central limit theorems for functionals of Gamma-correlated random fields with long-range dependence, deriving explicit limit distributions and their representations using advanced stochastic analysis techniques.

Contribution

It introduces a reduction theorem for Gamma-correlated fields and derives explicit limit distributions for non-linear functionals with Laguerre rank one or two.

Findings

Derived characteristic function of the Rosenblatt-type limit distribution.

Obtained multiple Wiener-Itô integral representations of the limits.

Constructed infinite series representations for the limit random variables.

Abstract

A reduction theorem is proved for functionals of Gamma-correlated random fields with long-range dependence in d-dimensional space. In the particular case of a non-linear function of a chi-squared random field with Laguerre rank equal to one, we apply the Karhunen-Lo\'eve expansion and the Fredholm determinant formula to obtain the characteristic function of its Rosenblatt-type limit distribution. When the Laguerre rank equals one and two, we obtain the multiple Wiener-It\^o stochastic integral representation of the limit distribution. In both cases, an infinite series representation in terms of independent random variables is constructed for the limit random variables.