Marcinkiewicz Law of Large Numbers for Outer-products of Heavy-tailed, Long-range Dependent Data (1501.02415v1)

Abstract: 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.