Integral limit theorems taking into account large deviations when Cramer's condition does not hold. I

Let $\xi_1,\dots,\xi_n$ be a sequence of independent equally distributed random variables with $\mathbf M\xi_n=0$. Throughout the paper it is supposed that the density function $p(x)$ of $\xi^n$ has the property
$$ p(x)\sim e^{-|x|^{1-\varepsilon}},\quad0<\varepsilon<1,\quad|x|\to\infty. $$
The problem we deal with is to describe the behaviour of the probability $\mathbf P\{\xi_1+\dots+\xi_n>x\}$ when $x$ tends to infinity so that $x>\sqrt n$.