Large deviations for sums of random variables of two types (Q2773626)

Let \(\{\xi_i\}_{i=1}^\infty\) and \(\{\tau_i\}_{i=1}^\infty\) be two independent sequences of i.i.d.\ random variables with finite expectation distributed as \(\xi\) and \(\tau\), respectively. Put \(S_n=\xi_1+\cdots+\xi_n\) and \(T_m=\tau_1+\cdots+\tau_m\). The asymptotic behavior of the probability \(\mathbf{P}\{S_n+T_m\geq x\}\) is studied as \(x\to\infty\). The following three distribution types for \(\xi\) and \(\tau\) are considered separately: (i) the distributions with tails regularly varying at infinity, (ii) the distributions with semi-exponential tails, and (iii) the distributions with exponentially decreasing tails.