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

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.