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

The Marcinkiewicz Strong Law, lim n→∞ 1 n 1 p n ∑ k=1 (Dk −D) = 0 a.s. with p ∈ (1, 2), is studied for outer products Dk = XkX T k , where {Xk}, {Xk} are both twosided (multivariate) linear processes (with coefficient matrices (Cl), (Cl) and i.i.d. zero-mean innovations {Ξ}, {Ξ}). Matrix sequences Cl and Cl can decay slowly enough (as |l| → ∞) that {Xk, Xk} have long-range dependence while {Dk} can have heavy tails. In particular, the heavy-tail and longrange-dependence phenomena for {Dk} 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.