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

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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


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