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Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process

Barrera, David
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1460652609

Abstract Details

2016, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
In this dissertation, we show that the Central Limit Theorem and the Invariance Principle for Discrete Fourier Transforms discovered by Peligrad and Wu can be extended to the quenched setting. We show that the random normalization introduced to extend these results is necessary and we discuss its meaning. We also show the validity of the quenched Invariance Principle for fixed frequencies under some conditions of weak dependence. In particular, we show that this result holds in the martingale case. The discussion needed for the proofs allows us to show some general facts apparently not noticed before in the theory of convergence in distribution. In particular, we show that in the case of separable metric spaces the set of test functions in the Portmanteau theorem can be reduced to a countable one, which implies that the notion of quenched convergence, given in terms of convergence a.s. of conditional expectations, specializes in the right way in the regular case when the state space is metrizable and second-countable. We also recover and organize several facts from the existing theory in a way consistent for the statistical spectral analysis of the Discrete Fourier Transforms, providing a comprehensive introduction to topics in this theory that apparently have not been systematically addressed in a self-contained way by previous references.
Magda Peligrad, Ph.D. (Committee Chair)
Wlodzimierz Bryc, Ph.D. (Committee Member)
Yizao Wang, Ph.D. (Committee Member)
145 p.
Mathematics
Quenched Convergence; Discrete Fourier Transforms; Central Limit Theorem; Invariance Principle

Recommended Citations

Citations

  • Barrera, D. (2016). Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1460652609

    APA Style (7th edition)

  • Barrera, David. Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process. 2016. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1460652609.

    MLA Style (8th edition)

  • Barrera, David. "Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process." Doctoral dissertation, University of Cincinnati, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1460652609

    Chicago Manual of Style (17th edition)

ucin1460652609
487
© 2016, some rights reserved.Creative Commons License Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process by David Barrera is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by University of Cincinnati and OhioLINK.