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

Let $\xi_1,\dots\xi_n,\dots$ be a sequence of independent equally distributed random variables and $\mathbf M\xi_n=0$. The density function $p(x)$ of $\xi_n$ being assumed to satisfy the condition
$$ p(x)\sim e^{-|x|^{1-\varepsilon}},\quad0<\varepsilon<1,\quad\text{as }|x|\to\infty, $$
the behaviour of the probability $\mathbf P\{\xi_i+\dots+\xi_n>x\}$ is studied when $n$ and $x$ tend to infinity so that $x>\sqrt n$.