Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

Let $\{\xi_i\}_{i=1}$ be a sequence of independent identically distributed nonnegative random variables, $S_n=\xi_1+\dots+\xi_n$. Let $\Delta=(0,T]$ and $x+\Delta=(x,x+T]$. We study the ratios of the probabilities $\mathbf{P}(s_n\in x+\Delta)/\mathbf{P}(\xi\in x+\Delta)$ for all $n$ and $x$. The estimates uniform in $x$ for these ratios are known for the so-called $\Delta$-subexponential distributions. Here we improve these estimates for two subclasses of $\Delta$-subexponential distributions; one of them is a generalization of the well-known class $\mathscr{SC}$ to the case of the interval $(0,T]$ with an arbitrary $T\leqslant\infty$. Also, a characterization of the class $\mathscr{SC}$ is given.