Concentration Inequalities for Statistical Inference

This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in awide range of settings,fromdistribution-free to distribution-dependent,from sub-Gaussian to sub-exponential,sub-Gamma,and sub-Weibull random variables,and from the mean to the maximum concentration.This review provides results in these settings with some fresh new results.Given the increasing popularity of high-dimensional data and inference,results in the context of high-dimensional linear and Poisson regressions are also provided.We aim to illustrate the concentration inequalities with known constants and to improve existing bounds with sharper constants.