Random Covering Sets in Metric Space with Exponentially Mixing Property

Let $\{B(\xi_n,r_n)\}_{n\ge1}$ be a sequence of random balls whose centers $\{\xi_n\}_{n\ge1}$ is a stationary process, and $\{r_n\}_{n\ge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=\limsup\limits_{n\to\infty}B(\xi_n,r_n)$, 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.