On approximation theorems for the Euler characteristic with applications to the bootstrap

Krebs, Johannes; Roycraft, Benjamin; Polonik, Wolfgang

doi:10.48550/arxiv.2005.07557

Cited by 1 publication

(2 citation statements)

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“…We anticipate that the same approach is possible for our Euler characteristic. A similar line of research in the non-extreme value theoretic setup can be found in [8,18].…”

Section: Introductionmentioning

confidence: 69%

“…In this paper, we first define a geometric complex that generalizes both R(X , t) and Č(X , t) above, and then we establish the functional strong law of large numbers (FSLLN) for the corresponding Euler characteristic. In conjunction with the recent development of Topological Data Analysis, the literature dealing with the asymptotics of the Euler characteristic of random geometric complexes has flourished [3,4,7,8,18]. However, none of these studies have paid sufficient attention to the topology of the tail of a probability distribution.…”

Section: Introductionmentioning

confidence: 99%

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Functional strong laws of large numbers for Euler characteristic processes of extreme sample clouds

Thomas

Owada

2020

Preprint

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This study demonstrates functional strong law large numbers for the Euler characteristic process of random geometric complexes formed by random points outside of an expanding ball in R d , in two distinct extreme value theoretic scenarios. When the points are drawn from a heavy-tailed distribution with a regularly varying tail, the Euler characteristic process grows at a regularly varying rate, and the scaled process converges uniformly and almost surely to a smooth function. When the points are drawn from a distribution with an exponentially decaying tail, the Euler characteristic process grows logarithmically, and the scaled process converges to another smooth function in the same sense. All of the limit theorems take place when the points inside the expanding ball are densely distributed, so that the simplex counts outside of the ball of all dimensions contribute to the Euler characteristic process.

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“…We anticipate that the same approach is possible for our Euler characteristic. A similar line of research in the non-extreme value theoretic setup can be found in [8,18].…”

Section: Introductionmentioning

confidence: 69%