Mathematics – Probability
Scientific paper
2006-06-13
Annals of Probability 2008, Vol. 36, No. 2, 739-764
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/07-AOP340 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
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 $\ell$ 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.
Jonasson Johan
Steif Jeffrey E.
No associations
LandOfFree
Dynamical models for circle covering: Brownian motion and Poisson updating does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Dynamical models for circle covering: Brownian motion and Poisson updating, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Dynamical models for circle covering: Brownian motion and Poisson updating will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-46098