## The Annals of Probability

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

### 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.

#### 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