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November, 1983 An Integral Test for the Rate of Escape of $d$-Dimensional Random Walk
Philip S. Griffin
Ann. Probab. 11(4): 953-961 (November, 1983). DOI: 10.1214/aop/1176993444


Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables taking values in $\mathbb{R}^d$ and $S_n = X_1 + \cdots + X_n$. For a large class of random variables, which includes all of those in the domain of attraction of a type $A$ stable law, an integral test is given which determines whether $P\{|S_n| \leq \gamma_n \mathrm{i.o.}\} = 0 \quad\text{or}\quad 1$ for any increasing sequence $\{\gamma_n\}$. This result generalizes the Dvoretzky-Erdos test for simple random walk and the Takeuchi and Taylor test for stable random walks.


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Philip S. Griffin. "An Integral Test for the Rate of Escape of $d$-Dimensional Random Walk." Ann. Probab. 11 (4) 953 - 961, November, 1983.


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

zbMATH: 0524.60068
MathSciNet: MR714958
Digital Object Identifier: 10.1214/aop/1176993444

Primary: 60J15
Secondary: 60F15

Keywords: domains of attraction , Integral test , probability estimates , Rate of escape

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 4 • November, 1983
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