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Limit Theorems for Random Fields

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2019, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
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.
Magda Peligrad, Ph.D. (Committee Chair)
Wlodzimierz Bryc, Ph.D. (Committee Member)
Yizao Wang, Ph.D. (Committee Member)
124 p.

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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)