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Limit theorems for functionals of linear processes in critical regions
Yudan Xiong, Fangjun Xu, Jinjiong Yu · 2025-02-28 · via math.PR updates on arXiv.org

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.

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