Multivariate limit theorems in the context of long-range dependence

TL;DR

This paper investigates the asymptotic behavior of vectors composed of normalized sums of functions of long-range dependent Gaussian series, revealing various possible limit laws depending on parameters and function ranks.

Contribution

It provides a comprehensive analysis of multivariate limit theorems for long-range dependent Gaussian processes, including new results for different Hermite ranks and a conjecture for general cases.

Findings

Limit laws can be Gaussian or Hermite-based depending on parameters.

The paper characterizes the limit processes for different Hermite ranks.

Includes a conjecture for the general case of Hermite ranks.

Abstract

We study the limit law of a vector made up of normalized sums of functions of long-range dependent stationary Gaussian series. Depending on the memory parameter of the Gaussian series and on the Hermite ranks of the functions, the resulting limit law may be (a) a multivariate Gaussian process involving dependent Brownian motion marginals, or (b) a multivariate process involving dependent Hermite processes as marginals, or (c) a combination. We treat cases (a), (b) in general and case (c) when the Hermite components involve ranks 1 and 2. We include a conjecture about case (c) when the Hermite ranks are arbitrary.