On almost sure limit theorems for heavy-tailed products of long-range dependent linear processes

Abstract

Marcinkiewicz strong law of large numbers, n−1p∑k=1n(dk−d)→0 almost surely with p∈(1,2), are developed for products dk=∏r=1sxk(r), where xk(r)=∑l=−∞∞ck−l(r)ξl(r) are two-sided linear processes with coefficients {cl(r)}l∈Z and i.i.d. zero-mean innovations {ξl(r)}l∈Z. The decay of the coefficients cl(r) as |l|→∞, can be slow enough for {xk(r)} to have long memory while {dk} can have heavy tails. The long-range dependence and heavy tails for {dk} are handled simultaneously and a decoupling property shows the convergence rate is dictated by the worst of long-range dependence and heavy tails, but not their combination. The Marcinkiewicz strong law of large numbers is also extended to the multivariate linear process case.