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
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
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.
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.
