  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.
