We study rates of convergence, in mean, for the Hausdorff metric between a finite set of stationary random variables and their common support, which is supposed to be a compact subset of $R^d$. We propose two different approaches for this study. The first approach is based on the notion of minimal index. This notion is introduced in this paper. It is in the spirit of the extremal index which is very used in the extreme value theory. The second approach is based on a $\beta$-mixing condition together with a local type dependence assumption. More precisely, all our results concern stationary $\beta$-mixing sequences satisfying a tail condition, known as the $(a,b)$-standard assumption, together with a local type dependence condition or stationary sequences satisfying the $(a,b)$-standard assumption and having a positive minimal index. We prove that the optimal rates of the i.i.d. setting can be reached. We apply our results to stationary Markov chains on a ball, or to a class of Markov chains on a circle or on a torus. We study with simulations the particular examples of a Möbius Markov chain on the unit circle and of a Markov chain on the unit square wrapped on a torus.