Open Access
August, 1977 Slowing Down $d$-Dimensional Random Walks
K. Bruce Erickson
Ann. Probab. 5(4): 645-651 (August, 1977). DOI: 10.1214/aop/1176995776


If $\{S_n\}$ is a genuinely $d$-dimensional random walk and $d \geqq 3$, then with probability 1, $n^{-\alpha}|S_n| \rightarrow \infty$ as $n \rightarrow \infty$ for every $\alpha < \frac{1}{2}$. This follows from a recent result of H. Kesten. In this paper we show that, under certain conditions, there is a constant $\alpha_0$ depending on the walk, but $\frac{1}{2} - 1/d \leqq \alpha_0 < \frac{1}{2}$, and a deterministic sequence of vectors $\{\nu_n\}$ such that $\lim \inf_n n^{-\alpha}|S_n - \nu_n| = 0$ with probability 1 for every $\alpha \geqq \alpha_0$. In discrete time this phenomenon cannot occur for any $\alpha < \frac{1}{2} - 1/d$; in continuous time it can occur for any $\alpha > 0$.


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K. Bruce Erickson. "Slowing Down $d$-Dimensional Random Walks." Ann. Probab. 5 (4) 645 - 651, August, 1977.


Published: August, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0369.60084
MathSciNet: MR458599
Digital Object Identifier: 10.1214/aop/1176995776

Primary: 60J15
Secondary: 60G50

Keywords: Random walk , Rate of escape

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 4 • August, 1977
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