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October, 1986 Answers to Some Questions About Increments of a Wiener Process
Chen Guijing, Kong Fanchao, Lin Zhengyan
Ann. Probab. 14(4): 1252-1261 (October, 1986). DOI: 10.1214/aop/1176992366

Abstract

Let $W(t), 0 \leq t < \infty$, be a Wiener process. This paper proves that $\lim \sup_{T \rightarrow \infty} \sup_{0 < t \leq T} \frac{|W(T) - W(T - t)|}{\{2t(\log(T/t) + \log\log t)\}^{1/2}} = 1, a.s.,$ $\lim_{T \rightarrow \infty} \sup_{0 < t \leq T} \sup_{t \leq s \leq T} \frac{|W(T) - W(s - t)|}{\{2t(\log(T/t) + \log \log t)\}^{1/2}} = 1, a.s.$ These results give an affirmative answer to the questions posed by Hanson and Russo without additional assumptions.

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Chen Guijing. Kong Fanchao. Lin Zhengyan. "Answers to Some Questions About Increments of a Wiener Process." Ann. Probab. 14 (4) 1252 - 1261, October, 1986. https://doi.org/10.1214/aop/1176992366

Information

Published: October, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0613.60027
MathSciNet: MR866346
Digital Object Identifier: 10.1214/aop/1176992366

Subjects:
Primary: 60F15
Secondary: 60G15 , 60G17

Keywords: Almost sure convergence , Increments of a Wiener process , Law of iterated logarithm , Wiener process

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 4 • October, 1986
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