Weak law of large numbers for linear processes

Abstract

We establish sufficient conditions for the Marcinkiewicz-Zygmund type weak law of large numbers for a linear process $\{X_k:k\in \mathbb{Z}\}$ defined by $X_k={\sum }_{j=0}^{\infty }{\psi }_j{\epsilon }_{k-j}$ for $k\in \mathbb{Z}$, where $\{{\psi }_j:j\in \mathbb{Z}\}\subset \mathbb{R}$ and $\{{\epsilon }_k:k\in \mathbb{Z}\}$ are independent and identically distributed random variables such that $x^p\mathrm{Pr}\{|{\epsilon }_0|>x\}\to 0$ as $x\to \infty$ with $1<p<2$ and $E{\epsilon }_0=0$. We use an abstract norming sequence that does not grow faster than $n^{1/p}$ if $\sum |{\psi }_j|<\infty$. If $\sum |{\psi }_j|=\infty$, the abstract norming sequence might grow faster than $n^{1/p}$ as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz-Zygmund type weak law of large numbers for the linear process.

17 pages