## The Annals of Probability

### 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

Michael J. Wichura

#### 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

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