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.
Publié le : 1976-08-14
Classification:
$r$-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,
60F99,
62L10
@article{1176996031,
author = {Lai, Tze Leung},
title = {On $r$-Quick Convergence and a Conjecture of Strassen},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 612-627},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996031}
}
Lai, Tze Leung. On $r$-Quick Convergence and a Conjecture of Strassen. Ann. Probab., Tome 4 (1976) no. 6, pp. 612-627. http://gdmltest.u-ga.fr/item/1176996031/