Large Deviations of Sums of Random Variables of Two Types

Let $\xi_1,\xi_2,\dots$; $\tau_1,\tau_2,\dots$ be two sequences of independent random variables, with $\xi_i$ and $\tau_i$ distributed respectively as $\xi$ and $\tau$ and with
$$ \mathbb{E}|\xi|<\infty, \quad \mathbb{E}|\tau|<\infty, \quad S_n=\sum_{i=1}^n\xi_i, \quad T_m=\sum_{i=1}^m\tau_i. $$
In this article we study the asymptotics of large deviation probabilities of the sums $T_m+S_n$ for the following three classes of distribution tails for $\tau$ and $\xi$: regular (heavy), semiexponential, and exponentially decreasing. The numbers $m$ and $n$ may be either fixed or unboundedly increasing. The cause for appearance of this article is the articles [1, 2] addressing a particular case of the problem under consideration.