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 $p\in(1,2)$, 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 $\{c_l^{(r)}\}_{l\in \mathbb{Z}}$ and i.i.d. zero-mean innovations $\{\xi_l^{(r)}\}_{l\in \mathbb{Z}}$. The decay of the coefficients $c_l^{(r)}$ as $|l|\to\infty$, can be slow enough for $\{x_k^{(r)}\}$ to have long memory while $\{d_k\}$ can have heavy tails. 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.

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