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Gaussian Approximations and Bootstrap with p >> n.

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We show that central limit theorems hold for high-dimensional normalized means hitting high dimensional rectangles (and, more generally, convex sets formed as sparse deformations of rectangles). These results apply even when p>> n. These theorems provide Gaussian distributional approximations that are not pivotal, but they can be consistently estimated via Gaussian multiplier methods (Gine and Zinn) and the empirical bootstrap. These results generalize to the suprema of empirical processes indexed by function sets with diverging complexity, and are useful for building confidence bands in modern high-dimensional and nonparametric problems (Gine and Nickl) and for multiple testing via the step-down methods. This is joint work with Denis Chetverikov (UCLA) and Kengo Kato (Tokyo). Refs: arxiv 1212.6906, 1212.6885,1303.7152.

This talk is part of the Probability Theory and Statistics in High and Infinite Dimensions series.

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