Open Access
March 2008 Dynamical models for circle covering: Brownian motion and Poisson updating
Johan Jonasson, Jeffrey E. Steif
Ann. Probab. 36(2): 739-764 (March 2008). DOI: 10.1214/07-AOP340


We consider two dynamical variants of Dvoretzky’s classical problem of random interval coverings of the unit circle, the latter having been completely solved by L. Shepp. In the first model, the centers of the intervals perform independent Brownian motions and in the second model, the positions of the intervals are updated according to independent Poisson processes where an interval of length is updated at rate α where α≥0 is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the nth interval is c/n, then there are times at which a fixed point is not covered if and only if c<2 and there are times at which the circle is not fully covered if and only if c<3. For the Poisson updating model, we obtain analogous results with c<α and c<α+1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.


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Johan Jonasson. Jeffrey E. Steif. "Dynamical models for circle covering: Brownian motion and Poisson updating." Ann. Probab. 36 (2) 739 - 764, March 2008.


Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1147.60063
MathSciNet: MR2393996
Digital Object Identifier: 10.1214/07-AOP340

Primary: 60K99

Keywords: Brownian motion , Circle coverings , Exceptional times , Hausdorff dimension

Rights: Copyright © 2008 Institute of Mathematical Statistics

Vol.36 • No. 2 • March 2008
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