High Dimensional Probability

Contents:

• Preliminaries on random variables. Classical inequalities and limit theorems.
• Concentration of sums of independent random variables: Hoeffding, Chernoff, and Bernstein inequalities, sub-Gaussian and sub-exponential distributions. Applications to random graphs.
• Random vectors in high dimension: concentration of the norm, covariance matrices and principal component analysis, high-dimensional distributions, sub-Gaussian distributions in higher dimensions. Applications: Grothendieck's Inequality, semidefinite programming, and maximum cut for graphs.
• Non-asymptotic analysis of random matrices: nets, covering and packing numbers, bounds on sub-Gaussian matrices, covariance estimation and clustering. Applications to error correcting codes, community detection in networks, and covariance estimation and clustering.
• Concentration of Lipschitz functions on the sphere, Johnson–Lindenstrauss theorem, matrix Bernstein inequality, community detection in sparse networks.
• Random processes: basic concepts, Slepian's inequality, bounds on Gaussian matrices, Sudakov's minoration inequality, Gaussian width, random projections of sets.
• Chaining: Dudley's inequality, empirical processes, Vapnik-Chervonenkis dimension with applications to statistical learning theory.