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Topological reconstruction of compact supports of dependent stationary random variables

Part of: Limit theorems Homotopy theory Stochastic processes Nonparametric inference

Published online by Cambridge University Press:  02 April 2024

Sadok Kallel  and
Sana Louhichi
Sadok Kallel*
Affiliation:
American University of Sharjah
Sana Louhichi*
Affiliation:
Université Grenoble Alpes
*
*Postal address: American University of Sharjah, UAE, and Laboratoire Painlevé, Université de Lille, France. Email address: skallel@aus.edu
**Postal address: Université Grenoble Alpes, CNRS, Grenoble INP, LJK 38000 Grenoble, France. Email address: sana.louhichi@univ-grenoble-alpes.fr
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Abstract

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.

Keywords

Hausdorff distancestationary dependent random variablesmixingMarkov chainscompact supportconcentrationpositive reachtopological inference

MSC classification

Primary: 60F99: None of the above, but in this section 60G10: Stationary processes
Secondary: 62G05: Estimation 55P10: Homotopy equivalences

Information

Type
Original Article
Information
Advances in Applied Probability , Volume 56 , Issue 4 , December 2024 , pp. 1339 - 1369
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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