## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 6 (1974), 1108-1138.

### Functional laws of the Iterated Logarithm for the Partial Sums of I. I. D. Random Variables in the Domain of Attraction of a Completely Asymmetric Stable Law

#### Abstract

Suppose $X$ and $X_n, n \geqq 1$, are i.i.d. random variables whose common distribution lies in the domain of attraction of a completely asymmetric stable law of index $\alpha (0 < \alpha < 2)$, so that (i) as $\nu \rightarrow \infty, \nu \rightarrow P\{X \geqq \nu\}$ varies regularly with exponent $-\alpha$, and (ii) $\lim_{\nu\rightarrow\infty} P\{X \leqq - \nu\}/P\{X \geqq \nu\} = 0$. Under a condition only slightly more strigent than (ii), we present Strassen-type functional laws of the iterated logarithm for the partial sums $S_n = \sum_{m\leqq n} X_m, n \geqq 1$. Our laws hold in particular when $X \geqq 0$; the proofs in this case utilize some new large deviation results for the $S_n$'s.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 6 (1974), 1108-1138.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996501

**Digital Object Identifier**

doi:10.1214/aop/1176996501

**Mathematical Reviews number (MathSciNet)**

MR358950

**Zentralblatt MATH identifier**

0325.60029

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G17: Sample path properties 60G50: Sums of independent random variables; random walks 60F10: Large deviations 60J30

**Keywords**

Functional law of the iterated logarithm completely asymmetric stable distribution large deviations

#### Citation

Wichura, Michael J. Functional laws of the Iterated Logarithm for the Partial Sums of I. I. D. Random Variables in the Domain of Attraction of a Completely Asymmetric Stable Law. Ann. Probab. 2 (1974), no. 6, 1108--1138. doi:10.1214/aop/1176996501. https://projecteuclid.org/euclid.aop/1176996501