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

Abstract

Let X₀, X₁,… be i.i.d. random variables with E(X₀) = 0, E(X²₀) = 1, E(exp{tX₀}) < ∞ (|t| < t₀) and partial sums S_n. Starting from Cramér's theorem on large deviations of the Cesàro means S_n/n, we show that other weighted sums V(n) = ∑a_j(n)X_j exhibit a similar large deviation behaviour, if the weights satisfy certain regularity conditions.