Limit theorems for linear processes with tapered innovations and filters

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).