Large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation

Abstract

Let {Xn, 1 ≤ n ≤ ∞} be a sequence of independent identically distributed random variables in the domain of attraction of a stable law with index 0 < α < 2. The limit of x-1 n log P{Sn/ max |Xi|≥ xn} is found when xn → ∞ and xn/n → 0. The large deviation result is used to prove the law of the iterated logarithm for the self-normalized partial sums.