On a lower bound of large – deviation probabilities for the sample mean under the Cramer condition

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. $\mathbf P(\bar X_n\ge x_n)\to0$.