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

TL;DR

This paper extends the Marcinkiewicz Strong Law of Large Numbers to outer products of heavy-tailed, long-range dependent multivariate linear processes, revealing that convergence is governed by the dominant tail or dependence.

Contribution

It introduces a new decoupling property for heavy-tailed, long-range dependent data and establishes convergence rates for outer products in this complex setting.

Findings

Convergence rate determined by the worst of heavy tails or long-range dependence.

Established Marcinkiewicz Strong Law for outer products of complex processes.

Applied results to stochastic approximation and autocovariance analysis.

Abstract

The Marcinkiewicz Strong Law, $\displaystyle\lim_{n\to\infty}\frac{1}{n^{\frac1p}}\sum_{k=1}^n (D_{k}- D)=0$ a.s. with $p\in(1,2)$, is studied for outer products $D_k=X_k\overline{X}_k^T$, where $\{X_k\},\{\overline{X}_k\}$ are both two-sided (multivariate) linear processes ( with coefficient matrices $(C_l), (\overline{C}_l)$ and i.i.d.\ zero-mean innovations $\{\Xi\}$, $\{\overline{\Xi}\}$). Matrix sequences $C_l$ and $\overline{C}_l$ can decay slowly enough (as $|l|\to\infty$) that $\{X_k,\overline{X}_k\}$ 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 Marcinkiewicz Strong Law of Large Numbers for stochastic approximation, non-linear functions forms and autocovariances.