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June 2016 Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data
Michael A. Kouritzin, Samira Sadeghi
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Adv. in Appl. Probab. 48(2): 349-368 (June 2016).

Abstract

The Marcinkiewicz strong law, limn→∞(1 / n1/p)∑k=1n(Dk - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products Dk = {XkX̅kT}, where {Xk} and {X̅k} are both two-sided (multivariate) linear processes (with coefficient matrices (Cl), (C̅l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences Cl and C ̅l can decay slowly enough (as |l| → ∞) that {Xk,X ̅k} 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 a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.

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Michael A. Kouritzin. Samira Sadeghi. "Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data." Adv. in Appl. Probab. 48 (2) 349 - 368, June 2016.

Information

Published: June 2016
First available in Project Euclid: 9 June 2016

zbMATH: 1342.62124
MathSciNet: MR3511765

Subjects:
Primary: 60F15 , 62J10 , 62J12
Secondary: 62L20

Keywords: Covariance , heavy tails , linear process , long-range dependence , Marcinkiewicz strong law of large numbers , stochastic approximation

Rights: Copyright © 2016 Applied Probability Trust

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Vol.48 • No. 2 • June 2016
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