Large Deviations for the Maximum of the Absolute Value of Partial Sums of Random Variable Sequences

Abstract

Let {ξi:i≥1} be a sequence of independent, identically distributed (i.i.d. for short) centered random variables. Let Sn=ξ1+⋯+ξn denote the partial sums of {ξi}. We show that sequence {1nmax1≤k≤n|Sk|:n≥1} satisfies the large deviation principle (LDP, for short) with a good rate function under the assumption that P(ξ1≥x) and P(ξ1≤−x) have the same exponential decrease.