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November, 1983 Some More Results on Increments of the Wiener Process
D. L. Hanson, Ralph P. Russo
Ann. Probab. 11(4): 1009-1015 (November, 1983). DOI: 10.1214/aop/1176993449


Let $W(T)$ for $0 \leq T < \infty$ be a standard Weiner process and suppose that $c_k$ and $b_k$ are fixed sequences of real numbers satisfying $0 \leq c_k < b_k < \infty$. Let $K(\omega)$ be the set of limit points (as $T \rightarrow \infty$) of $\frac{W(b_k;\omega) - W(c_k;\omega)}{\{2(b_k - c_k)\lbrack\log(b_k/(b_k - c_k)) + \log\log b_k\rbrack\}^{1/2}}$ where $\omega$ is a point in the probability space on which $W(T)$ is defined. We give necessary conditions on $b_k$ and $c_k$ to have $K(\omega) = \lbrack -1, 1\rbrack$ a.s. We also give some related results and discuss sharpness.


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D. L. Hanson. Ralph P. Russo. "Some More Results on Increments of the Wiener Process." Ann. Probab. 11 (4) 1009 - 1015, November, 1983.


Published: November, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0521.60033
MathSciNet: MR714963
Digital Object Identifier: 10.1214/aop/1176993449

Primary: 60F15
Secondary: 60G15 , 60G17

Keywords: delayed sums , Increments of a Wiener process , lag sums , Law of iterated logarithm , sums of random variables , Wiener process

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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