Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression
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Concentration inequalities form an essential toolkit in the study of high-dimensional statistical methods. Most of the relevant statistics literature is based on the assumptions of sub-Gaussian/sub-exponential random vectors. In this paper, we bring together various probability inequalities for sums of independent random variables under much weaker exponential type (sub-Weibull) tail assumptions. These results extract a part sub-Gaussian tail behavior in finite samples, matching the asymptotics governed by the central limit theorem, and are compactly represented in terms of a new Orlicz quasi-norm - the Generalized Bernstein-Orlicz norm - that typifies such tail behaviors. We illustrate the usefulness of these inequalities through the analysis of four fundamental problems in high-dimensional statistics. In the first two problems, we study the rate of convergence of the sample covariance matrix in terms of the maximum elementwise norm and the maximum k-sub-matrix operator norm which are key quantities of interest in bootstrap procedures and high-dimensional structured covariance matrix estimation. The third example concerns the restricted eigenvalue condition, required in high dimensional linear regression, which we verify for all sub-Weibull random vectors under only marginal (not joint) tail assumptions on the covariates. To our knowledge, this is the first unified result obtained in such generality. In the final example, we consider the Lasso estimator for linear regression and establish its rate of convergence under much weaker tail assumptions (on the errors as well as the covariates) than those in the existing literature. The common feature in all our results is that the convergence rates under most exponential tails match the usual ones under sub-Gaussian assumptions. Finally, we also establish a high-dimensional CLT and tail bounds for empirical processes for sub-Weibulls.
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