Limit theorems for functionals of linear processes in critical regions

TL;DR

This paper investigates the asymptotic behavior of partial sums of linear processes with heavy-tailed innovations in two critical parameter regions, filling gaps in existing limit theorem literature.

Contribution

It establishes limit theorems for the partial sums of linear processes in two previously unresolved critical parameter regions.

Findings

Identifies asymptotic distributions in critical region I: $eta=1$, $ ext{for } ext{α} ext{ in } (1,2)$.

Derives limit theorems for critical region II: $ ext{α}eta=2$, $eta ext{≥}1$.

Abstract

Let $X=\{X_n: n\in\mathbb{N}\}$ be the linear process defined by $X_n=\sum^{\infty}_{j=1} a_j\varepsilon_{n-j}$, where the coefficients $a_j=j^{-\beta}\ell(j)$ are constants with $\beta>0$ and $\ell$ a slowly varying function, and the innovations $\{\varepsilon_n\}_{n\in\mathbb{Z}}$ are i.i.d. random variables belonging to the domain of attraction of an $\alpha$-stable law with $\alpha\in(0,2]$. Limit theorems for the partial sum $ S_{[Nt]}=\sum^{[Nt]}_{n=1}[K(X_n)-\mathbb{E}K(X_n)]$ with proper measurable functions $K$ have been extensively studied, except for two critical regions: I. $\alpha\in(1,2),\beta=1$ and II. $\alpha\beta=2,\beta\geq1$. In this paper, we address these open scenarios and identify the asymptotic distributions of $S_{[Nt]}$ under mild conditions.