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Limit theory for U-statistics under geometric and topological constraints with rare events

Part of: Geometric probability and stochastic geometry Applied homological algebra and category theory Stochastic processes

Published online by Cambridge University Press:  14 September 2022

Takashi Owada
Takashi Owada*
Affiliation:
Purdue University
*
*Postal address: Department of Statistics, Purdue University, West Lafayette, 47907, USA. Email address: owada@purdue.edu
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Abstract

We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

Keywords

Point processpersistent Betti numberMorse critical pointextreme value theory

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 55U10: Simplicial sets and complexes 60G70: Extreme value theory; extremal processes
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 314 - 340
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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