Random Covering Sets in Metric Space with Exponentially Mixing Property

Abstract

Let $\{B({\xi }_n,r_n){\}}_{n\ge 1}$ be a sequence of random balls whose centers $\{{\xi }_n{\}}_{n\ge 1}$ is a stationary process, and $\{r_n{\}}_{n\ge 1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=\underset{n\to \infty }{\mathrm{limsup}}B({\xi }_n,r_n)$, 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.