Multivariate Limit Theorems in the Context of Long‐range Dependence

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 multi-variate Gaussian process involving dependent Brownian motion marginals, (b) a multi-variate process involving dependent Hermite processes as marginals or (c) a combination. We treat cases (a) and (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, although the conjecture can be resolved in some special cases.

Modeling temporally uncorrelated components of complex-valued stationary processes

“…There is a large literature on limit theorems under long-range dependence. See, e.g., [3,11,34] for a few central works on the topic. For an insightful and detailed presentation we refer to [18], and for recent advances on the topic, see, e.g., [21,40].…”

“…Similarly, for q ≥ 3 the limiting distribution is not Gaussian. For details on Hermite distributions and processes, we refer to [3,12,36]. In particular, we apply the following known result (see [11,Theorem 1]).…”

“…Indeed, if these conditions were violated, then we would need to study the convergence of complicated dependent random vectors towards some combinations of Hermite distributions. Unfortunately, to the best of our knowledge, there are no studies in this topic that would fit into our general setting (on some results in special cases, see, e.g., [3]).…”

“…converges to a nontrivial limit only if max i∈I q 2,i (2H i − 2) = q 1,j (4H j − 4). Finally, if both of these conditions are satisfied, the convergence in Equation ( 20) follows from [3,Theorem 4].…”

A Self‐Normalized Semi‐Parametric Test to Detect Changes in the Long Memory Parameter

“…Long memory processes , also known as strongly dependent processes or long‐range dependent processes , have attracted considerable attention in the literature; see for example recent monographs by Giraitis et al (), Beran et al () and Pipiras and Taqqu (). For long memory processes, an important quantity is the long memory parameter, which characterizes the degree of long‐range dependence and often appears in the asymptotic distribution of interested statistics; see for example Davydov (), Taqqu (), Dobrushin and Major (), Surgailis (), Avram and Taqqu (), Wu () and Bai and Taqqu () among many others. Due to its importance, extensive research has been conducted on the estimation of the long memory parameter; see for example Geweke and Porter‐Hudak (), Fox and Taqqu (), Künsch (), Dahlhaus (), Giraitis and Surgailis (), Robinson, (), Velasco, (), Phillips and Shimotsu (), Shimotsu and Phillips (), Shao and Wu () and references therein.…”