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##### REGULARITY - 2013
Overall Objectives
Application Domains
Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Dissemination
Bibliography
Overall Objectives
Application Domains
Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Dissemination
Bibliography

## 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 $\epsilon >0$,

 $|𝔼{\xi }_{i}^{k}|\le \frac{1}{2}k!{\epsilon }^{k-2}𝔼{\xi }_{i}^{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}k\ge 2\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}i=1,...,n.$ (12)

Denote by

 ${S}_{n}=\sum _{i=1}^{n}{\xi }_{i}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\sigma }^{2}=\sum _{i=1}^{n}𝔼{\xi }_{i}^{2}.$ (13)

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

 $\begin{array}{c}\hfill ℙ\left({S}_{n}>x\sigma \right)\phantom{\rule{4pt}{0ex}}\le \phantom{\rule{4pt}{0ex}}\underset{\lambda \ge 0}{inf}𝔼{e}^{\lambda \left({S}_{n}-x\sigma \right)}.\end{array}$ (14)

In the i.i.d. case, Cramér (1938) has established a large deviation expansion under the condition $𝔼{e}^{|{\xi }_{1}|}<\infty$. For all $0\le x=o\left(\sqrt{n}\right)$, one has

 $\frac{ℙ\left({S}_{n}>x\sigma \right)}{1-\Phi \left(x\right)}={e}^{\frac{{x}^{3}}{\sqrt{n}}\lambda \left(\frac{x}{\sqrt{n}}\right)}\left[1+O\left(\frac{1+x}{\sqrt{n}}\right)\right],\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n\to \infty ,$ (15)

where $\lambda \left(·\right)={c}_{1}+{c}_{2}\frac{x}{\sqrt{n}}+...$ 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

 $ℙ\left(\frac{{S}_{n}}{n}>y\right)=\frac{{inf}_{\lambda \ge 0}𝔼{e}^{\lambda \left({S}_{n}-yn\right)}}{{\sigma }_{y}{t}_{y}\sqrt{2\pi n}}\left[1+O\left(\frac{{c}_{y}}{n}\right)\right],\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n\to \infty ,$ (16)

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

 $\begin{array}{c}\hfill ℙ\left({S}_{n}>x\sigma \right)=\underset{\lambda \ge 0}{inf}𝔼{e}^{\lambda \left({S}_{n}-x\sigma \right)}F\left(x,\frac{\epsilon }{\sigma }\right),\end{array}$ (17)

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

 $\begin{array}{c}\hfill F\left(x,\frac{\epsilon }{\sigma }\right)=M\left(x\right)+28\phantom{\rule{0.166667em}{0ex}}\theta R\left(4x\epsilon /\sigma \right)\frac{\epsilon }{\sigma }\phantom{\rule{4pt}{0ex}}\end{array}$ (18)

with

 $\begin{array}{c}\hfill R\left(t\right)=\frac{{\left(1-t+6{t}^{2}\right)}^{3}}{{\left(1-3t\right)}^{3/2}{\left(1-t\right)}^{7}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0\le t<\frac{1}{3},\end{array}$ (19)

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

 $\left|ℙ\left({S}_{n}>x\sigma \right)-M\left(x\right)\underset{\lambda \ge 0}{inf}𝔼{e}^{\lambda \left({S}_{n}-x\sigma \right)}\right|=O\left(\frac{1}{\sqrt{n}}\underset{\lambda \ge 0}{inf}𝔼{e}^{\lambda \left({S}_{n}-x\sigma \right)}\right)$ (20)

and thus

 $\frac{ℙ\left({S}_{n}>x\sigma \right)}{M\left(x\right){inf}_{\lambda \ge 0}𝔼{e}^{\lambda \left({S}_{n}-x\sigma \right)}}=1+o\left(1\right).$ (21)