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December, 1974 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
Ann. Probab. 2(6): 1108-1138 (December, 1974). DOI: 10.1214/aop/1176996501

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.

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Michael J. Wichura. "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 (6) 1108 - 1138, December, 1974. https://doi.org/10.1214/aop/1176996501

Information

Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0325.60029
MathSciNet: MR358950
Digital Object Identifier: 10.1214/aop/1176996501

Subjects:
Primary: 60F15
Secondary: 60F10 , 60G17 , 60G50 , 60J30

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

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 6 • December, 1974
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