  Constant-specified and exponential concentration inequalities play an
essential role in the finite-sample theory of machine learning and
high-dimensional statistics area. We obtain sharper and constants-specified
concentration inequalities for the sum 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 the mean. These bounds are
new and improve existing bounds with sharper constants. In addition, a new
sub-Weibull parameter if the italic should be retained. Please check the whole
text. is also proposed, which enables recovering the tight concentration
inequality for a random variable (vector). For statistical applications, we
give an $\ell_2$-error of estimated coefficients in negative binomial
regressions when the heavy-tailed covariates are sub-Weibull distributed with
sparse structures, which is a new result for negative binomial regressions. In
applying random matrices, we derive non-asymptotic versions of Bai-Yin's
theorem for sub-Weibull entries with exponential tail bounds. Finally, by
demonstrating a sub-Weibull confidence region for a log-truncated Z-estimator
without the second-moment condition, we discuss and define the sub-Weibull type
robust estimator for independent observations $\{X_i\}_{i=1}^{n}$ without
exponential-moment conditions.
