On non-stationary Wishart matrices and functional Gaussian approximations in Hilbert spaces

This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.

Author dc:creator

• Dang, Thanh

Advisor dc:contributor.advisor

• Bourguin, Solesne

• Mathematics
• Gamma calculus
• Limit theorems
• Malliavin-Stein method
• Random matrices

Language dc:language.iso

en_US

Handle dc:identifier.uri

https://hdl.handle.net/2144/45267

OAI identifier oai:identifier

oai:open.bu.edu:2144/45267