The Annals of Probability

Dynamical models for circle covering: Brownian motion and Poisson updating

Johan Jonasson and Jeffrey E. Steif

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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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Ann. Probab., Volume 36, Number 2 (2008), 739-764.

First available in Project Euclid: 29 February 2008

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Primary: 60K99: None of the above, but in this section

Circle coverings Brownian motion exceptional times Hausdorff dimension


Jonasson, Johan; Steif, Jeffrey E. Dynamical models for circle covering: Brownian motion and Poisson updating. Ann. Probab. 36 (2008), no. 2, 739--764. doi:10.1214/07-AOP340.

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