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

Part of: Linear inference, regression Limit theorems Sequential methods

Published online by Cambridge University Press:  10 June 2016

Michael A. Kouritzin  and
Samira Sadeghi
Michael A. Kouritzin*
Affiliation:
University of Alberta
Samira Sadeghi*
Affiliation:
University of Alberta
*
* Postal address: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
* Postal address: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada.
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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.

Keywords

Covariancelinear processMarcinkiewicz strong law of large numbersheavy tailslong-range dependencestochastic approximation

MSC classification

Primary: 62J10: Analysis of variance and covariance 62J12: Generalized linear models 60F15: Strong theorems
Secondary: 62L20: Stochastic approximation
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 349 - 368
Copyright
Copyright © Applied Probability Trust 2016 

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