The Annals of Probability

Limit theorems for point processes under geometric constraints (and topological crackle)

Takashi Owada and Robert J. Adler

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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.

Article information

Source
Ann. Probab., Volume 45, Number 3 (2017), 2004-2055.

Dates
Received: March 2015
Revised: January 2016
First available in Project Euclid: 15 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1494835236

Digital Object Identifier
doi:10.1214/16-AOP1106

Mathematical Reviews number (MathSciNet)
MR3650420

Zentralblatt MATH identifier
1373.60090

Subjects
Primary: 60G70: Extreme value theory; extremal processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F05: Central limit and other weak theorems 55U10: Simplicial sets and complexes

Keywords
Point process Poisson random measure extreme value theory regular variation Čech complex Betti number topological data analysis crackle geometric graph

Citation

Owada, Takashi; Adler, Robert J. Limit theorems for point processes under geometric constraints (and topological crackle). Ann. Probab. 45 (2017), no. 3, 2004--2055. doi:10.1214/16-AOP1106. https://projecteuclid.org/euclid.aop/1494835236


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