A Large Deviation Principle for Weighted Sums of Independent Identically Distributed Random Variables

Let X0 X1 be i.i.d. random variables with E X0 = 0, E X2 0 = 1, E exp tX0 < ∞ |t| < t0 and partial sums Sn. Starting from Cram´er’s theorem on large deviations of the Ces`aro means Sn/n, we show that other weighted sums V n = aj n Xj exhibit a similar large deviation behaviour, if the weights satisfy certain regularity conditions.