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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.20956 |
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| _version_ | 1866915176763097088 |
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| author | Xiong, Yudan Xu, Fangjun Yu, Jinjiong |
| author_facet | Xiong, Yudan Xu, Fangjun Yu, Jinjiong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limit theorems for functionals of linear processes in critical regions Xiong, Yudan Xu, Fangjun Yu, Jinjiong Probability 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. |
| title | Limit theorems for functionals of linear processes in critical regions |
| topic | Probability |
| url | https://arxiv.org/abs/2502.20956 |