The Annals of Probability

On $r$-Quick Convergence and a Conjecture of Strassen

Tze Leung Lai

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Abstract

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.

Article information

Source
Ann. Probab., Volume 4, Number 4 (1976), 612-627.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996031

Digital Object Identifier
doi:10.1214/aop/1176996031

Mathematical Reviews number (MathSciNet)
MR431326

Zentralblatt MATH identifier
0369.60036

JSTOR
links.jstor.org

Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 62L10: Sequential analysis

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

Citation

Lai, Tze Leung. On $r$-Quick Convergence and a Conjecture of Strassen. Ann. Probab. 4 (1976), no. 4, 612--627. doi:10.1214/aop/1176996031. https://projecteuclid.org/euclid.aop/1176996031


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