Let ${B(xi_n,r_n)}_{nge1}$ be a sequence of random balls whose centers ${xi_n}_{nge1}$ is a stationary process, and ${r_n}_{nge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=limsuplimits_{ntoinfty}B(xi_n,r_n)$, that is, the points covered by $B(xi_n,r_n)$ infinitely often. The sizes of $E$ are investigated from the viewpoint of measure, dimension and topology.
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