##
Xiequan Fan

Sharp Large deviations for sums of independent random variables

### Résumé

We obtain some optimal inequalities on tail probabilities for sums of
independent random variables. For bounded summands, the results complete
an upper bound on tail probabilities due to Talagrand. For non bounded
summands, we improve Bennett's inequality under Bernstein's condition in
the spirit of Hoeffding and Talagrand. In particular, we obtain a
one-term asymptotic expansion for large deviations, which can be
regarded as sharp large deviation result of types of Cramér and
Bahadur-Rao. This asymptotic expansion improves the classical standard
of Bernstein.