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...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.展开更多
Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116： 27-37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting...Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116： 27-37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability （W-UNI） of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W- UNI, one of which is a W-UNI analogue of the celebrated de La Vall6e Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.1127116161300204)
文摘In this paper, the Kolmogorov-Feller type weak law of large numbers are obtained, which extend and improve the related known works in the literature.
基金funded by National Natural Science Foundation of China(Grants 92046021,12071013,12026607,71973005)LMEQF at Peking University.
文摘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.
文摘Recently, T. K. Chandra, T. -C. Hu and A. Rosalsky [Statist. Probab. Lett., 2016, 116： 27-37] introduced the notion of a sequence of random variables being uniformly nonintegrable, and presented a list of interesting results on this uniform nonintegrability. We introduce a weaker definition on uniform nonintegrability （W-UNI） of random variables, present a necessary and sufficient condition for W-UNI, and give two equivalent characterizations of W- UNI, one of which is a W-UNI analogue of the celebrated de La Vall6e Poussin criterion for uniform integrability. In addition, we give some remarks, one of which gives a negative answer to the open problem raised by Chandra et al.
