An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions

We obtain an integro-local limit theorem for the sum $S(n)=\xi(1)+\cdots+\xi(n)$ of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form $\mathbf P(\xi\ge t)=t^{-\beta}L(t)$ with $\beta>2$ and some slowly varying function $L(t)$. The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities
$$\mathbf P(S(n)\in[x,x+\Delta))$$
as $x\to\infty$ for a fixed $\Delta>0$; i.e., in the domain where the normal approximation applies, in the domain where $S(n)$ is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains.