## The Annals of Probability

### Some "LIM INF" Results for Increments of a Wiener Process

#### Abstract

Let $W(t)$ for $0 \leq t < \infty$ be a standard Wiener process, suppose $0 < a_T \leq T$ for $T > 0$, and let $d(T, t) = \{2t\lbrack\log(T/t) + \log \log t \rbrack\}^{1/2}$. Quantities such as $\lim, \inf_{T \rightarrow \infty} \sup_{a_T \leq t \leq T} \frac{W(T) - W(T - t)}{d(T,t)},$ $\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq t \leq T - a_T\\0 \leq s \leq a_T}} \frac{|W(t + s) - W(t)|}{d(t + a_T, a_T)}$ and $\lim, \inf_{T \rightarrow \infty} \sup_{\substack{0 \leq u < v \leq T\\a_T \leq v - u}} \frac{|W(v) - W(u)|}{d(v, v - u)}$ are investigated.

#### Article information

Source
Ann. Probab., Volume 17, Number 3 (1989), 1063-1082.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991257

Digital Object Identifier
doi:10.1214/aop/1176991257

Mathematical Reviews number (MathSciNet)
MR1009445

Zentralblatt MATH identifier
0684.60021

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G15: Gaussian processes 60G17: Sample path properties

#### Citation

Hanson, D. L.; Russo, Ralph P. Some "LIM INF" Results for Increments of a Wiener Process. Ann. Probab. 17 (1989), no. 3, 1063--1082. doi:10.1214/aop/1176991257. https://projecteuclid.org/euclid.aop/1176991257