Asymptotic Behavior of Large Deviation Probabilities for Two Weighted Sums of Random Variables

We consider two weighted sums of independent identically distributed nonlattice variables. We assume that the mean of the first sum is less than the mean of the second sum and consider the probability of the rare event that the first sum is greater than the second one. We assume Cramer’s condition for the summands. Under some additional assumptions we study the asymptotic behavior of the probability above. The results are applied to the gladiator model introduced by Kaminsky, Luks, and Nelson.