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August, 1976 On $r$-Quick Convergence and a Conjecture of Strassen
Tze Leung Lai
Ann. Probab. 4(4): 612-627 (August, 1976). DOI: 10.1214/aop/1176996031


In this paper, we prove a conjecture of Strassen on the set of $r$-quick limit points of the normalized linearly interpolated sample sum process in $C\lbrack 0, 1 \rbrack$. We give the best possible moment conditions for this conjecture to hold by finding the $r$-quick analogue of the classical law of the iterated logarithm and its converse. The proof is based on an $r$-quick version of Strassen's strong invariance principle and a theorem on the $r$-quick limit set of a semi-stable Gaussian process. Some applications of Strassen's conjecture are given. We also consider the notion of $r$-quick convergence related to the law of large numbers and outline some statistical applications to indicate the usefulness of this concept.


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Tze Leung Lai. "On $r$-Quick Convergence and a Conjecture of Strassen." Ann. Probab. 4 (4) 612 - 627, August, 1976.


Published: August, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0369.60036
MathSciNet: MR431326
Digital Object Identifier: 10.1214/aop/1176996031

Primary: 60F99
Secondary: 62L10

Keywords: $r$-quick convergence , $r$-quick limit points , last time , Law of the iterated logarithm , Marcinkiewicz-Zygmund strong law , sample sums , semi-stable Gaussian process , sequential analysis , Strassen's conjecture , Strong invariance principle

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 4 • August, 1976
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