  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.

Probability