Geodesic
On a lower bound of large -- deviation probabilities for the sample mean under the Cramer condition
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 4, Tome 278 (2001), pp. 208-224

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Let $X_1,X_2,\dots$ be i.i.d. random variables, satisfying the condition $$ \mathbf EX_1^2 e^{\lambda X_1}\infty\ (\exists\,\lambda>0). $$ We investigate the asymptotic behavior of $\mathbf P(\bar X_n\ge x)$ as $n\to\infty$ provided that $\bar X_n=\frac{X_1+\dots+X_n}{n}$, when $x\ge x_n>\mathbf EX_1$ and $x_n$ is such that $\bar X_n$ is contained in a zone of large deviations, i.e. $\mathbf P(\bar X_n\ge x_n)\to0$. 
@article{ZNSL_2001_278_a12,
     author = {L. V. Rozovskii},
     title = {On a lower bound of large -- deviation probabilities for the sample mean under the {Cramer} condition},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {208--224},
     publisher = {mathdoc},
     volume = {278},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_278_a12/}
}
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L. V. Rozovskii. On a lower bound of large -- deviation probabilities for the sample mean under the Cramer condition. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 4, Tome 278 (2001), pp. 208-224. http://geodesic.mathdoc.fr/item/ZNSL_2001_278_a12/