Limit theorems for functionals of linear processes in critical regions

Abstract

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