Limit theorems for linear processes with tapered innovations and filters

TL;DR

This paper investigates the asymptotic behavior of partial sums of linear processes with tapered filters and heavy-tailed innovations, revealing Gaussian limits depending on tapering growth rates and dependence structures.

Contribution

It introduces a comprehensive analysis of limit theorems for linear processes with both tapered filters and innovations, extending understanding of their asymptotic Gaussian behavior.

Findings

Limit processes are Gaussian under various tapering growth scenarios.

The asymptotic behavior depends on the growth rates of tapering parameters.

Different dependence structures influence the convergence to Gaussian limits.

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 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 case where $b(n)$ grows relatively slow (soft tapering), and all three cases of growth of $\l(n)$ (strong, weak, and moderate tapering). In these cases the limit processes (in the sense of convergence of finite dimensional distributions) are Gaussian.