Dynamical models for circle covering: Brownian motion and Poisson updating
成果类型：
Article
署名作者：
Jonasson, Johan; Steif, Jeffrey E.
署名单位：
Chalmers University of Technology; University of Gothenburg
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/07-AOP340
发表日期：
2008
页码：
739-764
关键词：
摘要：
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 C is updated at rate l(-a) where alpha >= 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 < alpha and c < alpha + 1 instead. We also compute the Hausdorff dimension of the set of exceptional times for some of these questions.