MARCINKIEWICZ LAW OF LARGE NUMBERS FOR OUTER PRODUCTS OF HEAVY-TAILED, LONG-RANGE DEPENDENT DATA

The Marcinkiewicz strong law, lim(n ->infinity)(1/n(1/p)) Sigma(n)(k= 1)(D-k - D) = 0 almost surely with p is an element of(1, 2), is studied for outer products D-k = XkX (X) over bar (inverted perpendicular)(k) , where {X-k} and {(X) over bar (k)} are both two-sided (multivariate) linear processes (with coefficient matrices (C-l), ((C) over bar (l)) and independent and identically distributed zero-mean innovations {Xi} and {Xi}). Matrix sequences C-l and (C) over bar (l) can decay slowly enough (as vertical bar l vertical bar -> infinity) that {X-k, (X) over bar (k)} have longrange dependence, while {D-k} can have heavy tails. In particular, the heavy-tail and longrange-dependence phenomena for {D-k} are handled simultaneously and a newdecoupling 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.