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

الأصل · EN

The Marcinkiewicz Strong Law, ₙ→∞1n1p∑ₖ₌₁ⁿ (Dₖ- D)=0 a.s. with p∈(1,2), is studied for outer products Dₖ=XₖXₖᵗ, where {Xₖ},{Xₖ} are both two-sided (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 long-range-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.