The purpose of this paper is to establish a connection between stochastic extreme value theory and certain aspects of topological data analysis. We propose two assumptions analogous to Leadbetter’s conditions (known as Conditions \(\varvec{D(u_n)}\) and \(\varvec{D'(u_n)}\) ), which are widely used in extreme value theory. Under these assumptions, we derive rates of convergence, in expectation, for the Hausdorff metric between a finite set of stationary dependent random variables and their common support, assumed to be a compact subset of \(\mathbb {R}^{\varvec{d}}\) . We show that the optimal rate established by Chazal et al. (J. Mach. Learn. Res. 16, 3603–3635 2015) in the i.i.d. case is reached. Our results apply, for instance, to a class of compactly supported stationary Markov chains, as well as to stationary \(\varvec{\varphi }\) -mixing, \(\varvec{\rho }\) -mixing, or \(\varvec{\alpha }\) -mixing sequences.
