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May 2017 Limit theorems for point processes under geometric constraints (and topological crackle)
Takashi Owada, Robert J. Adler
Ann. Probab. 45(3): 2004-2055 (May 2017). DOI: 10.1214/16-AOP1106

Abstract

We study the asymptotic nature of geometric structures formed from a point cloud of observations of (generally heavy tailed) distributions in a Euclidean space of dimension greater than one. A typical example is given by the Betti numbers of Čech complexes built over the cloud. The structure of dependence and sparcity (away from the origin) generated by these distributions leads to limit laws expressible via nonhomogeneous, random, Poisson measures. The parametrisation of the limits depends on both the tail decay rate of the observations and the particular geometric constraint being considered.

The main theorems of the paper generate a new class of results in the well established theory of extreme values, while their applications are of significance for the fledgling area of rigorous results in topological data analysis. In particular, they provide a broad theory for the empirically well-known phenomenon of homological “crackle;” the continued presence of spurious homology in samples of topological structures, despite increased sample size.

Citation

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Takashi Owada. Robert J. Adler. "Limit theorems for point processes under geometric constraints (and topological crackle)." Ann. Probab. 45 (3) 2004 - 2055, May 2017. https://doi.org/10.1214/16-AOP1106

Information

Received: 1 March 2015; Revised: 1 January 2016; Published: May 2017
First available in Project Euclid: 15 May 2017

zbMATH: 1373.60090
MathSciNet: MR3650420
Digital Object Identifier: 10.1214/16-AOP1106

Subjects:
Primary: 60G70, 60K35
Secondary: 55U10, 60D05, 60F05, 60G55

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 3 • May 2017
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