Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Suppose that $\xi,\xi(1),\xi(2),\dots$ are independent identically distributed random variables such that $-\xi$ is semiexponential; i.e., $\mathbf P(-\xi\geqslant t)=e^{-t^{\beta}L(t)}$, $\beta\in(0,1)$, $L(t)$ is a slowly varying function as $t\to\infty$ possessing some smoothness properties. Let $\mathbf E\xi=0$, $\mathbf D\xi=1$, and $S(k)=\xi(1)+\dots+\xi(k)$. Given $d>0$, define the first upcrossing time $\eta+(u)=\inf\{k\geqslant1:S(k)+kd>u\}$ at nonnegative level $u\geqslant0$ of the walk $S(k)+kd$ with positive drift $\d>0$. We prove that, under general conditions, the following relation is valid for $n\to\infty$ and for $u=u(n)\in[0,dn-N_n\sqrt{n}]$:
\begin{equation*} \mathbf P(\eta_+(u)>n)\thicksim\frac{\mathbf E_{\eta_+}(u)}{n}\mathbf P(S(n)\leqslant x), \tag{0.1} \end{equation*}
where $x=u-nd<0$ and an arbitrary fixed sequence $N_n$ not exceeding $d\sqrt{n}$ tends to $\infty$.
The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability $\mathbf P(S(n)\leqslant x)$ for $x\leqslant-\sqrt{n}$ (for $x\in[-\sqrt{n},0]$ it follows from the central limit theorem).