Sharp large deviation results for sums of independent random variables

Xiequan Fan (1, 2) , Ion Grama (3) , Quansheng Liu (3)

1 Regularity - Probabilistic modelling of irregularity and application to uncertainties management
2 MAS - Mathématiques Appliquées aux Systèmes - EA 4037
3 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

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

Keywords: Hoeffding's inequality; Bernstein's inequality; Bennett's inequality; sharp large deviations; expansion of Bahadur-Rao; Cramer large deviations; sums of independent random variables

Domains: Probability [math.PR]

Xiequan Fan, Ion Grama, Quansheng Liu. Sharp large deviation results for sums of independent random variables. Science China Mathematics, 2015, 58 (9), pp.1939-1958. ⟨10.1007/s11425-015-5049-6⟩. ⟨hal-01187587⟩