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Sharp rates of convergence in the Hausdorff metric for some compactly supported stationary sequences

Part of: Limit theorems Nonparametric inference Stochastic processes

Published online by Cambridge University Press:  17 June 2025

Sana Louhichi Open the ORCID record for Sana Louhichi [Opens in a new window]
Sana Louhichi*
Affiliation:
Université Grenoble Alpes
*
*Postal address: Université Grenoble Alpes, CNRS, Grenoble INP, LJK 38000 Grenoble, France. 700 Avenue Centrale, 38401 Saint-Martin-d’Hères, France. Email: sana.louhichi@univ-grenoble-alpes.fr
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Abstract

We study convergence rates, 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 $\mathbb{R}^d$. We propose two different approaches for this study. The first is based on the notion of a minimal index. This notion is introduced in this paper. It is in the spirit of the extremal index, which is much used in 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 independent and identically distributed 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.

Keywords

Hausdorff metricstationary dependent random variablesβ-mixingMöbius Markov chainMarkov chain on a torusgeometrically ergodic Markov chaincompact supportrates of convergenceextremal indexlocal dependence conditionBonferroni-type inequality

MSC classification

Primary: 60F99: None of the above, but in this section 60G10: Stationary processes
Secondary: 62G05: Estimation

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 4 , December 2025 , pp. 1446 - 1474
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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