Weak law of large numbers for linear processes

TL;DR

This paper establishes conditions under which a linear process satisfies a Marcinkiewicz-Zygmund type weak law of large numbers, extending classical results to processes with dependent structure and heavy-tailed innovations.

Contribution

It provides new sufficient conditions and norming sequences for the weak law of large numbers in linear processes with heavy-tailed innovations, including cases with infinite coefficient sums.

Findings

Weak law holds under certain summability conditions on coefficients

Norming sequence growth depends on the sum of coefficients

Rate of convergence varies with the structure of the linear process

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 $X_k=\sum_{j=0}^\infty\psi_j\varepsilon_{k-j}$ for $k\in\mathbb Z$, where $\{\psi_j:j\in\mathbb Z\}\subset\mathbb R$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed random variables such that $x^p\Pr\{|\varepsilon_0|>x\}\to0$ as $x\to\infty$ with $1<p<2$ and $\operatorname E\varepsilon_0=0$. 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.