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October, 1976 Recurrence Sets of Normed Random Walk in $R^d$
K. Bruce Erickson
Ann. Probab. 4(5): 802-828 (October, 1976). DOI: 10.1214/aop/1176995985


In this paper we give examples of the sets of recurrent points (or accumulation points) of random walks in $R^d$ normed by nice sequences of constants. These examples, interesting in their own right, give rise to some very interesting conjectures concerning the general structure of such sets. Of particular interest are the recurrent points of the ordinary averages or sample means. It turns out that any closed subset of $R^d$ can be the finite points of recurrence of a sequence of averages: $(X_1 + \cdots + X_n)/n, X_i$ i.i.d. random vectors. This seems to be a property not shared by most other normalizing sequences. We also give some results on rates of escape of random walks in a domain of attraction. In looking for rates of escape we are looking for normalizing constants which give rise to no finite recurrent points of the normalized walk.


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K. Bruce Erickson. "Recurrence Sets of Normed Random Walk in $R^d$." Ann. Probab. 4 (5) 802 - 828, October, 1976.


Published: October, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0362.60074
MathSciNet: MR426162
Digital Object Identifier: 10.1214/aop/1176995985

Primary: 60G50
Secondary: 60F05 , 60F15 , 60J15

Keywords: accumulation points , limit points , Normed sums of independent random variables , Random walk

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 5 • October, 1976
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