Advances in Applied Probability

Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data

Michael A. Kouritzin and Samira Sadeghi

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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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Adv. in Appl. Probab., Volume 48, Number 2 (2016), 349-368.

First available in Project Euclid: 9 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J10: Analysis of variance and covariance 62J12: Generalized linear models 60F15: Strong theorems
Secondary: 62L20: Stochastic approximation

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


Kouritzin, Michael A.; Sadeghi, Samira. Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data. Adv. in Appl. Probab. 48 (2016), no. 2, 349--368.

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