Large-Deviation Probabilities for Maxima of Sums of Independent Random Variables with Negative Mean and Subexponential Distribution

We consider the sums $S_n=\xi_1+\dots+\xi_n$ of independent identically distributed random variables with negative mean value. In the case of subexponential distribution of the summands, the asymptotic behavior is found for the probability of the event that the maximum of sums $\max(S_1,\ldots,S_n)$ exceeds high level $x$. The asymptotics obtained describe this tail probability uniformly with respect to all values of $n$.