Limit theorems for linear processes with tapered innovations and filters

Pub Date : 2024-02-16 DOI:10.1007/s10986-024-09619-1

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Abstract

We consider the partial-sum process \({\sum }_{k=1}^{\left[nt\right]}{X}_{k}^{\left(n\right)},\) where \(\left\{{X}_{k}^{\left(n\right)}={\sum }_{j=0}^{\infty }{\alpha }_{j}^{\left(n\right)}{\xi }_{k-j}\left(b\left(n\right)\right), k\in {\mathbb{Z}}\right\},\) n ≥ 1, is a series of linear processes with tapered filter \({\alpha }_{j}^{\left(n\right)}={\alpha }_{j} {1}_{\left\{0\le j\le\lambda\left(n\right)\right\}}\) and heavy-tailed tapered innovations ξ_j(b(n)), j ∈ Z. Both tapering parameters b(n) and ⋋ (n) grow to ∞ as n→∞. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter a_i, i ≥ 0, and nontapered innovations. We consider the cases where b(n) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of ⋋(n) (strong, weak, and moderate tapering).

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具有锥形创新和滤波器的线性过程的极限定理

Abstract We consider the partial-sum process \({\sum }_{k=1}^{left[nt\right]}{X}_{k}^{left(n\right)}、\其中 \(left\{X}_{k}^{left(n\right)}={sum }_{j=0}^{infty } {alpha }_{j}^{left(n\right)}{xi }_{k-j}\left(b\left(n\right)\right)、kin {\mathbb{Z}}\right\},\) n ≥ 1、是一系列线性过程，具有锥形滤波器 \({\alpha }_{j}^{left(n\right)}={\alpha }_{j} {1}_{\left\{0le jle\lambda\left(n\right)\right}}) 和重尾锥形创新 ξj(b(n)), j∈ Z。当 n→∞ 时，锥形参数 b(n) 和 ⋋ (n) 都增长到 ∞。偏和过程的极限行为（在有限维分布收敛的意义上）取决于这两个渐减参数的增长，以及具有非渐减滤波 ai、i ≥ 0 和非渐减创新的线性过程的依赖特性。我们考虑了 b(n)增长相对较慢（软渐缩）和较快（硬渐缩）的情况，以及⋋(n)增长的所有三种情况（强渐缩、弱渐缩和适度渐缩）。

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