Skip to Main Content
Frequently Asked Questions
Submit an ETD
Global Search Box
Need Help?
Keyword Search
Participating Institutions
Advanced Search
School Logo
Files
File List
33098.pdf (990.19 KB)
ETD Abstract Container
Abstract Header
Limit Theorems for Random Fields
Author Info
Zhang, Na
ORCID® Identifier
http://orcid.org/0000-0001-6492-0402
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1563527352284677
Abstract Details
Year and Degree
2019, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
Abstract
The focus of this dissertation is on the dependent structure and limit theorems of high dimensional probability theory. In this dissertation, we investigate two related topics: the Central Limit Theorem (CLT) for stationary random fields (multi-indexed random variables) and for Fourier transform of stationary random fields. We show that the CLT for stationary random processes under a sharp projective condition introduced by Maxwell and Woodroofe in 2000 \cite{MW00} can be extended to random fields. To prove this result, new theorems are established herein for triangular arrays of martingale differences which have interest in themselves. Later on, to exploit the richness of martingale techniques, we establish the necessary and sufficient conditions for martingale approximation of random fields, which extend to random fields many corresponding results for random sequences (e.g. \cite{DMV07}). Besides, a stronger form of convergence, the quenched convergence, is investigated and a quenched CLT is obtained under some projective criteria. The discrete Fourier transform of random fields, $(X_{\bm{k}})_{\bm{k}\in\mathbb{Z}^d}$ $(d\geq 2)$, where $\mathbb{Z}$ is the set of integers, is defined as the rotated sum of the random fields. Being one of the important tools to prove the CLT for Fourier transform of random fields, the law of large numbers (LLN) is obtained for discrete Fourier transform of random sequences under a very mild regularity condition. Then the central limit theorem is studied for Fourier transform of random fields, where the dependence structure is general and no restriction is assumed on the rate of convergence to zero of the covariances.
Committee
Magda Peligrad, Ph.D. (Committee Chair)
Wlodzimierz Bryc, Ph.D. (Committee Member)
Yizao Wang, Ph.D. (Committee Member)
Pages
124 p.
Subject Headings
Mathematics
Keywords
central limit theorem
;
quenched central limit theorem
;
random fields
;
Fourier transform
;
projective condition
;
martingale approximation
Recommended Citations
Refworks
EndNote
RIS
Mendeley
Citations
Zhang, N. (2019).
Limit Theorems for Random Fields
[Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1563527352284677
APA Style (7th edition)
Zhang, Na.
Limit Theorems for Random Fields.
2019. University of Cincinnati, Doctoral dissertation.
OhioLINK Electronic Theses and Dissertations Center
, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1563527352284677.
MLA Style (8th edition)
Zhang, Na. "Limit Theorems for Random Fields." Doctoral dissertation, University of Cincinnati, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1563527352284677
Chicago Manual of Style (17th edition)
Abstract Footer
Document number:
ucin1563527352284677
Download Count:
348
Copyright Info
© 2019, some rights reserved.
Limit Theorems for Random Fields by Na Zhang 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.