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Outer Products and Stochastic Approximation Algorithms in a Heavy-tailed and Long-range Dependent Setting Open Access


Other title
Stochastic approximation
Linear process
Heavy tails
Marcinkiewicz strong law of large numbers
Long-range dependence
Rates of convergence
Type of item
Degree grantor
University of Alberta
Author or creator
Supervisor and department
Kouritzin, Michael (Mathematical and Statistical Sciences)
Examining committee member and department
Kouritzin, Michael (Mathematical and Statistical Sciences)
Christoph Frei (Mathematical and Statistical Sciences)
Byron Schmuland (Mathematical and Statistical Sciences)
Mark Lewis (Mathematical and Statistical Sciences)
Lajos Horvath (Department of Mathematics of Utah University)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Doctor of Philosophy
Degree level
Classical time-series theories are mainly concerned with the statistical analysis of light-tailed and short-range dependent stationary linear processes. Applications in network theory and financial mathematics lead us to consider time series models with heavy tails and long memory. Heavy-tailed data exhibits frequent extremes and infinite variance, while positively-correlated long memory data displays great serial momentum or inertia. Heavy-tailed data with long-range dependence has been observed in a plethora of empirical data set over the last fifty years and so. Methodological and theoretical results as well as a considerable portion of applied work in this thesis address long-range dependence and heavy-tailed types of the data. The first contribution of this thesis is the development of Marcinkiewicz strong law of large numbers for outer products of multivariate linear processes while handling long-range dependent and heavy-tailed data structure. This result is used to obtain Marcinkiewicz strong law of large numbers for non-linear function of partial sums, sample auto-covariances and linear processes in a stochastic approximation setting. The next part of the result is on developing almost sure convergence rates for linear stochastic approximation algorithms under some assumptions that are implied by Marcinkiewicz strong law of large numbers. Finally, we verify our results experimentally in the stochastic approximation setting while handling all gains, long-range dependence and heavy tails and addressing the optimal polynomial rate of convergence by establishing results akin to the Marcinkiewicz strong law of large numbers.
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
Citation for previous publication
Kouritzin, M.A. and Sadeghi, S. (2015). Convergence Rates and Decoupling in Linear Stochastic Approximation Algorithms. SIAM Journal on Control and Optimization, vol. 53-3, pp. 1484-1508.Kouritzin, M.A. and Sadeghi, S. (2015). Marcinkiewicz Law of Large Numbers for Outer-products of Heavy-tailed, Long-range-Dependence Data. Advances in Applied Probability Journal, in press.

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