Functional limit theorems for linear processes with tapered innovations and filters

TL;DR

This paper establishes functional limit theorems for linear processes with tapered filters and heavy-tailed innovations, analyzing how different tapering growth rates influence the convergence behavior of partial sum processes.

Contribution

It introduces a comprehensive framework for understanding the asymptotic behavior of linear processes with tapered filters and innovations, covering various tapering growth scenarios.

Findings

Limit behavior depends on tapering growth rates and dependence structure.

Different regimes of tapering (soft, hard, moderate) lead to distinct limit processes.

Results unify and extend existing limit theorems for linear processes with heavy tails.

Abstract

In the paper we consider the partial sum process $\sum_{k=1}^{[nt]}X_k^{(n)}$, where $\{X_k^{(n)}=\sum_{j=0}^{\infty} a_{j}^{(n)}\xi_{k-j}(b(n)), \ k\in \bz\},\ n\ge 1,$ is a series of linear processes with tapered filter $a_{j}^{(n)}=a_{j}\ind{[0\le j\le \l(n)]}$ and heavy-tailed tapered innovations $\xi_{j}(b(n), \ j\in \bz$. Both tapering parameters $b(n)$ and $\l(n)$ grow to $\infty$ as $n\to \infty$. 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 non-tapered filter $a_i, \ i\ge 0$ and non-tapered innovations. We consider the cases where $b(n)$ grows relatively slow (soft tapering) and rapidly (hard tapering), and all three cases of growth of $\l(n)$ (strong, weak, and moderate tapering).