Sharp large deviation results for sums of independent random variables  

作　　者：FAN XieQuan GRAMA Ion LIU QuanSheng 

机构地区：[1]Regularity Team, Institut National de Recherche en Informatique et en Automatique [2]Laboratoire de Mathmatiques Appliques aux Systèmes, Ecole Centrale Paris-Grande Voie des Vignes [3]Laboratoire de Mathmatiques de Bretagne Atlantique, UMR 6205, University Bretagne-Sud [4]School of Mathematics and Computing Science, Changsha University of Science and Technology

出　　处：《Science China Mathematics》2015年第9期1939-1958,共20页中国科学：数学（英文版）

基　　金：supported by the Post-Graduate Study Abroad Program sponsored by China Scholarship Council;National Natural Science Foundation of China(Grant Nos.11171044 and11401590)

摘　　要：We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand(1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cram′er(1938),Bahadur-Rao(1960) and Sakhanenko(1991). We also show that our bound can be used to improve a recent inequality of Pinelis(2014).We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein’s condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand（1995）. We also complete Talagrand’s inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cram′er（1938）,Bahadur-Rao（1960） and Sakhanenko（1991）. We also show that our bound can be used to improve a recent inequality of Pinelis（2014）.

关 键 词：Bernstein’s inequality sharp large deviations Cramér large deviations expansion of BahadurRao sums of independent random variabl 

分 类 号：O212.1[理学—概率论与数理统计]