Leadbetter-type conditions for bounding the Hausdorff metric of compactly supported stationary sequences
Résumé
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 $D(u_n )$ and $D′ (u_n )$), 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 $R^d$ . We show that the optimal rates established by Chazal et al. (2015) in the i.i.d. case are reached. Our results apply, for instance, to a class of compactly supported stationary Markov chains, as well as to stationary φ-mixing, ρ-mixing, or α-mixing sequences.
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