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

  • Author / Creator
  • 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.

  • Subjects / Keywords
  • Graduation date
    Fall 2015
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
  • Institution
    University of Alberta
  • Degree level
  • Department
  • Specialization
    • Statistics
  • Supervisor / co-supervisor and their department(s)
  • Examining committee members and their departments
    • Christoph Frei (Mathematical and Statistical Sciences)
    • Byron Schmuland (Mathematical and Statistical Sciences)
    • Mark Lewis (Mathematical and Statistical Sciences)
    • Kouritzin, Michael (Mathematical and Statistical Sciences)
    • Lajos Horvath (Department of Mathematics of Utah University)