Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data

Abstract

The Marcinkiewicz strong law, lim_n→∞(1 / n^1/p)∑_k=1ⁿ(D_k - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products D_k = {X_kX̅_k^T}, where {X_k} and {X̅_k} are both two-sided (multivariate) linear processes (with coefficient matrices (C_l), (C̅_l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences C_l and C ̅_l can decay slowly enough (as |l| → ∞) that {X_k,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 a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.