Integro-local and integral theorems for sums of random variables with semiexponential distributions

We obtain some integro-local and integral limit theorems for the sums $S(n)=\xi(1)+\dots+\xi(n)$ of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form $\mathbf P(\xi\geqslant t)=e^{-t^{\beta}L(t)}$, where $\beta\in(0,1)$ and $L(t)$ is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as $x\to\infty$ of the probabilities
$$ \mathbf P(S(n)\in[x,x+\Delta))\textrm{ and }\mathbf P(S(n)\geqslant x) $$
in the zone of normal deviations and all zones of large deviations of $x$: in the Cramer and intermediate zones, and also in the “extreme” zone where the distribution of $S(n)$ is approximated by that of the maximal summand.