On almost sure limit theorems for long-range dependent, heavy-tailed processes

Classical methods of inference are often rendered inapplicable while dealing with data exhibiting heavy tails, which gives rise to infinite variance and frequent extremes, and long memory, which induces inertia in the data. In this paper, we develop the Marcinkiewicz strong law of large numbers, ${n^{-\frac1p}}\sum_{k=1}^{n} (d_{k}- d)\rightarrow 0\$ almost surely with $p\in(1,2)$, for products $d_k=\prod_{r=1}^s x_k^{(r)}$, where each $x_k^{(r)} = \sum_{l=-\infty}^{\infty}c_{k-l}^{(r)}\xi_l^{(r)}$ is a two-sided univariate linear process with coefficients $\{c_l^{(r)}\}_{l\in \mathbb{Z}}$ and i.i.d. zero-mean innovations $\{\xi_l^{(r)}\}_{l\in \mathbb{Z}}$ respectively. The decay of the coefficients $c_l^{(r)}$ as $|l|\to\infty$, can be slow enough that $\{x_k^{(r)}\}$ can have long memory while $\{d_k\}$ can have heavy tails. The aim of this paper is to handle the long-range dependence and heavy tails for $\{d_k\}$ simultaneously, and to prove a decoupling property that shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The multivariate linear process case is also considered.