Taking as our point of departure the methods and results in a 1960 paper of Bahadur and Ranga Rao, we derive asymptotic representations of large deviation probabilities for weighted sums of independent, identically distributed random variables. The main theorem generalizes the Bahadur-Ranga Rao result in the absolutely continuous case. The method of proof closely parallels that of the 1960 paper, a major component of which was the use of Cramer's 1923 theorem on asymptotic expansions. For our result, we need an extension of Cramer's theorem to triangular arrays, and that extension is also developed in the paper. We then show that the main theorem implies a logarithmic result which generalizes a 1952 theorem of Chernoff and is of more precision but less generality than a 1969 result of Feller. Finally, we note that in the exponential case the theorem can be used to estimate large deviation probabilities for linear combinations of exponential order statistics.
Stephen A. Book. "Large Deviation Probabilities for Weighted Sums." Ann. Math. Statist. 43 (4) 1221 - 1234, August, 1972. https://doi.org/10.1214/aoms/1177692474