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In this paper we extend results on reconstruction of probabilistic supports of independent and identically distributed random variables to supports of dependent stationary 
${\mathbb R}^d$-valued random variables. All supports are assumed to be compact of positive reach in Euclidean space. Our main results involve the study of the convergence in the Hausdorff sense of a cloud of stationary dependent random vectors to their common support. A novel topological reconstruction result is stated, and a number of illustrative examples are presented. The example of the Möbius Markov chain on the circle is treated at the end with simulations.
Permutation products and their various “fat diagonal” subspaces are studied from the topological and geometric points of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We exhibit an expression for the fundamental group of any permutation product of a connected space 
$X$ having the homotopy type of a CW complex in terms of 
${{\pi }_{1}}(X)$ and 
${{H}_{1}}(X;\,\mathbb{Z})$. We then prove that the fundamental group of the configuration space of 
$n$-points on $X$, of which multiplicities do not exceed 
$n/2$, coincides with ${{H}_{1}}(X;\,\mathbb{Z})$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.
