  The Marcinkiewicz Strong Law,
$\displaystyle\lim_{n\to\infty}\frac{1}{n^{\frac1p}}\sum_{k=1}^n (D_{k}- D)=0$
a.s. with $p\in(1,2)$, is studied for outer products $D_k=X_k\overline{X}_k^T$,
where $\{X_k\},\{\overline{X}_k\}$ are both two-sided (multivariate) linear
processes ( with coefficient matrices $(C_l), (\overline{C}_l)$ and i.i.d.\
zero-mean innovations $\{\Xi\}$, $\{\overline{\Xi}\}$). Matrix sequences $C_l$
and $\overline{C}_l$ can decay slowly enough (as $|l|\to\infty$) that
$\{X_k,\overline{X}_k\}$ have long-range dependence while $\{D_k\}$ can have
heavy tails. In particular, the heavy-tail and long-range-dependence phenomena
for $\{D_k\}$ are handled simultaneously and a new decoupling property is
proved that shows the convergence rate is determined by the worst of the
heavy-tails or the long-range dependence, but not the combination. The main
result is applied to obtain Marcinkiewicz Strong Law of Large Numbers for
stochastic approximation, non-linear functions forms and autocovariances.
