Section: New Results
Large Deviations Inequalities
Participant : Xiequan Fan.
Let
Denote by
The well-known Bernstein inequality (1946) states that, for all
In the i.i.d. case, Cramér (1938) has established a large deviation expansion under the condition
where
Bahadur-Rao (1960) proved the following sharp large deviations similar to (15 ). Assume Cramér's condition. Then, for given
where
We present an improvement on Bernstein's inequality. In particular, we establish a sharp large deviation expansion similar to the classical results of Cramér and Bahadur-Rao. The following theorem is our main result.
Theorem 0.1 Assume Bernstein's condition. Then, for all
where
with
and
and thus