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    Convergence Rates of Spectral Distribution of Random Inner Product Kernel Matrices

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    TETDEDXKong-temple-0225E-13280.pdf
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    Genre
    Thesis/Dissertation
    Date
    2018
    Author
    Kong, Nayeong
    Advisor
    Rider, Brian (Brian C.)
    Committee member
    Yang, Wei-shih, 1954-
    Berhanu, Shiferaw
    Mukhopadhyay, Subhadeep
    Department
    Mathematics
    Subject
    Mathematics
    Probability
    Random Geometric Graph
    Random Graph
    Random Inner Product Kernel Matrix
    Random Matrix
    Spectral Distribution
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/3132
    
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    DOI
    http://dx.doi.org/10.34944/dspace/3114
    Abstract
    This dissertation has two parts. In the first part, we focus on random inner product kernel matrices. Under various assumptions, many authors have proved that the limiting empirical spectral distribution (ESD) of such matrices A converges to the Marchenko- Pastur distribution. Here, we establish the corresponding rate of convergence. The strategy is as follows. First, we show that for z = u + iv ∈ C, v > 0, the distance between the Stieltjes transform m_A (z) of ESD of matrix A and Machenko-Pastur distribution m(z) is of order O (log n \ nv). Next, we prove the Kolmogorov distance between ESD of matrix A and Marchenko-Pastur distribution is of order O(3\log n\n). It is the less sharp rate for much more general class of matrices. This uses a Berry-Esseen type bound that has been employed for similar purposes for other families of random matrices. In the second part, random geometric graphs on the unit sphere are considered. Observing that adjacency matrices of these graphs can be thought of as random inner product matrices, we are able to use an idea of Cheng-Singer to establish the limiting for the ESD of these adjacency matrices.
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