Dynamical models for circle covering: Brownian motion and Poisson updating
Abstract
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 $\ell$ is updated at rate $\ell^{\alpha}$ where $\alpha \ge0$ is a parameter. For the model with Brownian motions, a special case of our results is that if the length of the $n$th 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 $c<\alpha$ and $c<\alpha+1$ instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606297
 Bibcode:
 2006math......6297J
 Keywords:

 Mathematics  Probability;
 60K99 (Primary)
 EPrint:
 Published in at http://dx.doi.org/10.1214/07AOP340 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)