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

Abstract

Let ${X_n, 1 \leq n < \infty}$ be a sequence of independent identically distributed random variables in the domain of attraction of a stable law with index $0 < \alpha < 2$. The limit of $x_n^{-1}\log P{S_n/ \max |X_i| \geq x_n}$ is found when $x_n \to \infty$ and $\x_n/n \to 0$. The large deviation result is used to prove the law of the iterated logarithm for the self-normalized partial sums.