Section: New Results

Large Deviations Inequalities

Participant : Xiequan Fan.

Let ${\left({\xi }_i\right)}_{i=1,...,n}$ be a sequence of independent and centered random variables satisfying Bernstein's condition, for a constant $\epsilon >0$,

Denote by

The well-known Bernstein inequality (1946) states that, for all $x>0$,

In the i.i.d. case, Cramér (1938) has established a large deviation expansion under the condition $𝔼e^{|{\xi }_1|}<\infty$. For all $0\le x=o\left(\sqrt{n}\right)$, one has

where $\lambda (·)=c_1+c_2\frac{x}{\sqrt{n}}+...$ is the Cramér series and the values $c_1,c_2,...$ depend on the distribution of ${\xi }_1$.

Bahadur-Rao (1960) proved the following sharp large deviations similar to (15 ). Assume Cramér's condition. Then, for given $y>0$, there is a constant $c_y$ depending on the distribution of ${\xi }_1$ and $y$ such that

where $t_y$, ${\sigma }_y$ and $c_y$ depend on the distribution of ${\xi }_1$ and $y$.

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 $0\le x<\frac{1}{12}\frac{\sigma }{\epsilon }$,

where $\sqrt{2\pi }M\left(x\right)$ is the Mills ratio, the function

with

and $\left|\theta \right|\le 1$. In particular, in the i.i.d. case, for all $0\le x=o\left(\sqrt{n}\right),n\to \infty ,$

and thus