Sharper Sub-Weibull Concentrations: Non-asymptotic Bai-Yin Theorem
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Arising in high-dimensional probability, non-asymptotic concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics. In this article, we obtain a sharper and constants-specified concentration inequality for the summation of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations (from mean). These bounds improve existing bounds with sharper constants. In the application of random matrices, we derive non-asymptotic versions of Bai-Yin's theorem for sub-Weibull entries and it extends the previous result in terms of sub-Gaussian entries. In the application of negative binomial regressions, we gives the ℓ_2-error of the estimated coefficients when covariate vector X is sub-Weibull distributed with sparse structures, which is a new result for negative binomial regressions.
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