## The Annals of Probability

### How Big are the Increments of a Wiener Process?

#### Abstract

Let $\beta_T = (2a_T\lbrack\log(T/a_T) + \log\log T\rbrack)^{-\frac{1}{2}}, 0 < a_T \leqslant T < \infty$ and $\{W(t); 0 \leqslant t < \infty\}$ be a standard Wiener process. This paper studies the almost sure limiting behaviour of $\sup_{0\leqslant t\leqslant T-a_T} \beta_T|W(t + a_T) - W(t)|$ as $T \rightarrow \infty$ under varying conditions on $a_T = c \log T, c > 0$, the Erdos-Renyi law of large numbers for the Wiener process. A number of other results for the Wiener process also follow via choosing $a_T$ appropriately. Connections with strong invariance principles and the P. Levy modulus of continuity for $W(t)$ are also established.

#### Article information

Source
Ann. Probab. Volume 7, Number 4 (1979), 731-737.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176994994

Digital Object Identifier
doi:10.1214/aop/1176994994

Mathematical Reviews number (MathSciNet)
MR537218

JSTOR