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August, 1979 How Big are the Increments of a Wiener Process?
M. Csorgo, P. Revesz
Ann. Probab. 7(4): 731-737 (August, 1979). DOI: 10.1214/aop/1176994994


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.


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M. Csorgo. P. Revesz. "How Big are the Increments of a Wiener Process?." Ann. Probab. 7 (4) 731 - 737, August, 1979.


Published: August, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0412.60038
MathSciNet: MR537218
Digital Object Identifier: 10.1214/aop/1176994994

Primary: 60F15
Secondary: 60G15 , 60G17

Keywords: Increments of a Wiener process , Law of iterated logarithm

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 4 • August, 1979
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