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

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.