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

Marcinkiewicz strong law of large numbers, ${n^{-frac1p}}sum_{k=1}^{n} (d_{k}- d)rightarrow 0 $ almost surely with $pin(1,2)$, are developed for products $d_k=prod_{r=1}^s x_k^{(r)}$, where the $x_k^{(r)} = sum_{l=-infty}^{infty}c_{k-l}^{(r)}xi_l^{(r)}$ are two-sided linear process with coefficients ${c_l^{(r)}}_{lin mathbb{Z}}$ and i.i.d. zero-mean innovations ${xi_l^{(r)}}_{lin mathbb{Z}}$. The decay of the coefficients $c_l^{(r)}$ as $|l|toinfty$, can be slow enough for ${x_k^{(r)}}$ to have long memory while ${d_k}$ can have heavy tails. The long-range dependence and heavy tails for ${d_k}$ are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The results provide a means to estimate how much (if any) long-range dependence and heavy tails a sequential data set possesses, which is done for real financial data. All of the stocks we considered had some degree of heavy tails. The majority also had long-range dependence. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.