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This paper gives a review of concentration inequalities which are widely employed in non-asymptotical analyses of mathematical statistics in a wide range of settings, from distribution-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.
We explore the applications of our previously established likelihood-ratio method for deriving concentration inequalities for a wide variety of univariate and multivariate distributions. New concentration inequalities for various distributions are de
This paper presents compact notations for concentration inequalities and convenient results to streamline probabilistic analysis. The new expressions describe the typical sizes and tails of random variables, allowing for simple operations without hea
In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Frechet mean of some distribution $mathbb{P}$ supported on a subspace of positive semi-definite Hermitian operators $mathbb{H}_{+}(d)$. We allow a bary
This paper considers distributed statistical inference for general symmetric statistics %that encompasses the U-statistics and the M-estimators in the context of massive data where the data can be stored at multiple platforms in different locations.
We derive simple concentration inequalities for bounded random vectors, which generalize Hoeffdings inequalities for bounded scalar random variables. As applications, we apply the general results to multinomial and Dirichlet distributions to obtain multivariate concentration inequalities.