Limit theorems for functionals of linear processes in critical regions
Yudan Xiong, Fangjun Xu, Jinjiong Yu
TL;DR
This work resolves two open critical regions for limit theorems of functionals of linear processes with coefficients $a_j=j^{-\beta}\ell(j)$ and heavy-tailed innovations. It develops region-specific approximation schemes—a long-memory Fourier-type expansion $T_N$ and a short-memory truncation $S_{N,l}$—to obtain sharp asymptotic distributions for $S_{[Nt]}$ across regimes. In Region I ($α\in(1,2)$, $β=1$) the limit is an $α$-stable process with a nontrivial scaling $A_N$, while Region II ($αβ=2$) yields Gaussian limits with distinct normalizations, including a special case $α=2,β=1$ with $K'_{\infty}(0)\neq0$ producing a Brownian limit with variance determined by slowly varying components. When short memory applies on the critical curve, a standard Brownian limit with finite variance is recovered. The results rely on Fourier analysis, Karamata representations, and careful control of truncation and linearization errors, extending the theory to these previously unresolved critical regions.
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 $a_j=j^{-β}\ell(j)$ are constants with $β>0$ 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 $α$-stable law with $α\in(0,2]$. 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. $α\in(1,2),β=1$ and II. $αβ=2,β\geq1$. In this paper, we address these open scenarios and identify the asymptotic distributions of $S_{[Nt]}$ under mild conditions.