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.
Categories: math.ST stat.TH