Sharp large deviation results for sums of independent random variables

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• 作者：XieQuan Fan ; Ion Grama ; QuanSheng Liu
• 关键词：Bernstein’s inequality ; sharp large deviations ; Cramér large deviations ; expansion of Bahadur ; Rao ; sums of independent random variables ; Bennett’s inequality ; Hoeffding’s inequality ; 60F10 ; 60F05 ; 60E15 ; 60G50
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2015
• 出版时间：September 2015
• 年：2015
• 卷：58
• 期：9
• 页码：1939-1958
• 全文大小：366 KB
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• 作者单位：XieQuan Fan (1) (2)
  Ion Grama (3)
  QuanSheng Liu (3) (4)
  
  1. Regularity Team, Institut National de Recherche en Informatique et en Automatique, Palaiseau, 91120, France
  2. Laboratoire de Mathématiques Appliquées aux Systèmes, Ecole Centrale Paris-Grande Voie des Vignes, Chatenay-Malabry, 92295, France
  3. Laboratoire de Mathématiques de Bretagne Atlantique, UMR 6205, University Bretagne-Sud, Vannes, 56000, France
  4. School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, 410114, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

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 Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that our bound can be used to improve a recent inequality of Pinelis (2014).