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Dynamical models for circle covering: Brownian motion and Poisson updating. (English) Zbl 1147.60063
(Authors’ summary slightly altered) The authors 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 [Isr. J. Math. 11, 328–345 (1972; Zbl 0241.60008)]. In the first model, the centers of the intervals perform independent Brownian motions while in the second one, the positions of the intervals are updated according to independent Poisson processes where an interval of length $$\ell$$ is updated at rate $$\ell ^{-\alpha }$$, with $$\alpha >0$$ a parameter. For the model with Brownian motions, a special case of the authors’ 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, analogous results with $$c<\alpha$$ and $$c<\alpha +1$$ instead are obtained. The Hausdorff dimension of the set of exceptional times for some of these questions is also computed.

##### MSC:
 60K99 Special processes
Full Text:
##### References:
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