## The Annals of Probability

- Ann. Probab.
- Volume 24, Number 3 (1996), 1368-1387.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 24, Number 3 (1996), 1368-1387.

**Dates**

First available in Project Euclid: 9 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1065725185

**Digital Object Identifier**

doi:10.1214/aop/1065725185

**Mathematical Reviews number (MathSciNet)**

MR1411498

**Zentralblatt MATH identifier**

0869.60025

**Subjects**

Primary: 60F10: Large deviations 60F15: Strong theorems

Secondary: 60G50: Sums of independent random variables; random walks 60G18: Self-similar processes

**Keywords**

Stable law domain of attraction large deviation law of the iterated logarithm self-normalized partial sums largest absolute observation

#### Citation

Horváth, Lajos; Shao, Qi-Man. Large deviations and law of the iterated logarithm for partial sums normalized by the largest absolute observation. Ann. Probab. 24 (1996), no. 3, 1368--1387. doi:10.1214/aop/1065725185. https://projecteuclid.org/euclid.aop/1065725185