Large deviations of sums of random variables of two types (Q1876423)

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 \(\tau\) with \(\mathbb E | \xi| <\infty\), \(\mathbb E | \tau| <\infty\), \(S_n=\sum_{i=1}^n\xi_i\), \(T_m=\sum_{i=1}^m\tau_i\). The author studies 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. A particular case of the problem under consideration was studied in the articles by \textit{S. Asmussen, C. Klüppelberg,} and \textit{K. Sigman} [Stochastic Processes Appl. 79, No. 2, 265--286 (1999; Zbl 0961.60080)] and \textit{S. Foss} and \textit{D. Korshunov} [Markov Process. Relat. Fields 6, No. 4, 543--568 (2000; Zbl 0977.60091)].