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Marcinkiewicz strong law of large numbers, ${n^{-\frac1p}}\sum_{k=1}^{n}(d_{k}- d)\rightarrow 0\ $ almost surely with $p\in(1,2)$, are developed forproducts $d_k=\prod_{r=1}^s x_k^{(r)}$, where the $x_k^{(r)} =\sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}\xi_l^{(r)}$ are two-sided linearprocesses with coefficients $\{c_l^{(r)}\}_{l\in \mathbb{Z}}$ and i.i.d.zero-mean innovations $\{\xi_l^{(r)}\}_{l\in \mathbb{Z}}$. The decay of thecoefficients $c_l^{(r)}$ as $|l|\to\infty$, can be slow enough for$\{x_k^{(r)}\}$ to have long memory while $\{d_k\}$ can have heavy tails. Thelong-range dependence and heavy tails for $\{d_k\}$ are handled simultaneouslyand a decoupling property shows the convergence rate is dictated by the worstof long-range dependence and heavy tails, but not their combination. TheMarcinkiewicz strong law of large numbers is also extended to the multivariatelinear process case.

On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes