Convergence Rates of Spectral Distribution of Random Inner Product Kernel Matrices
Genre
Thesis/DissertationDate
2018Author
Kong, NayeongAdvisor
Rider, Brian (Brian C.)Committee member
Yang, Wei-shih, 1954-Berhanu, Shiferaw
Mukhopadhyay, Subhadeep
Department
MathematicsSubject
MathematicsProbability
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
Metadata
Show full item recordDOI
http://dx.doi.org/10.34944/dspace/3114Abstract
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.ADA compliance
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