Betti Numbers of Gaussian Excursions in the Sparse Regime

Thoppe, Gugan; Krishnan, Sunder Ram

doi:10.48550/arxiv.1807.11018

Cited by 2 publications

(2 citation statements)

References 28 publications

(78 reference statements)

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“…Various concentration bounds have also been established for N ⋆ (R, ℓ) [38,39,10], but these do not lead to improved bounds on the variance in the short-range correlated case. Related questions have also been studied in the 'sparse' regime ℓ R → ∞ as R → ∞ [42].…”

Section: Introductionmentioning

confidence: 99%

A central limit theorem for the number of excursion set components of Gaussian fields

Beliaev¹,

McAuley²,

Muirhead³

2022

Preprint

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For a smooth stationary Gaussian field on R d and level ℓ ∈ R, we consider the number of connected components of the excursion set {f ≥ ℓ} (or level set {f = ℓ}) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g. the Bargmann-Fock field), a central limit theorem holds with volume-order scaling. Previously such a result had only been established for 'additive' geometric functionals of the excursion/level sets (e.g. the volume or Euler characteristic) using Hermite expansions. Our approach, based on a martingale analysis, is more robust and can be generalised to a wider class of topological functionals. A major ingredient in the proof is a third moment bound on critical points, which is of independent interest.

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Section: Introductionmentioning

confidence: 99%

A central limit theorem for the number of excursion set components of Gaussian fields

Beliaev¹,

McAuley²,

Muirhead³

2022

Preprint

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“…Note that as a corollary, Theorem 2.2.2 also implies Poisson approximation for the finite-dimensional marginals of (A d,t ) t≤1 . Moreover, a Poisson approximation result with a diverging Poisson parameter can also be a way to derive a CLT [76,Theorem 3.10]. It appears that such an approach could establish a CLT in a regime where ρ d ↑ ∞ and lim sup d↑∞ ρ…”

Section: Model and Main Resultsmentioning

confidence: 99%

Limit theory for spatial random networks

Willhalm

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Samenvatting viAs the complexity and size of networks relevant for TDA increases, the need for a better comprehension of their behavior on a large scale becomes more and more important. Spatial networks can be modeled by arranging nodes randomly in a metric space and establishing connections based on spatial constraints or probabilities. The generated object is referred to as a spatial random network. This thesis aims to contribute to the understanding of spatial random networks by studying their limit theory. A major focus of this thesis is the analysis of rare events. Relatively small deviations from what is expected are the territory of central limit theorems. Rarer events can be approached with Poisson approximations or the theory of large deviations. The latter is the main focus of this thesis. vii Large deviationsThe theory of large deviations is a branch of probability theory that aims to quantify the probability of rare events. It provides powerful tools to study the asymptotic behavior of stochastic processes and has applications in a wide range xi More details about Poisson approximations can also be found in [8]. A first use case of the Poisson approximation in the context of rare events analysis goes back to [18], studying, among other things, the number of soldiers in the Prussian army killed by horse kicks.The Poisson approximation can be extended to a Poisson process approximation as outlined for instance in [23]. A Poisson process approximation in this spirit is derived in Chapter 2. For this, we need a notion of distance between two point xiii

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Betti Numbers of Gaussian Excursions in the Sparse Regime

Cited by 2 publications

References 28 publications

A central limit theorem for the number of excursion set components of Gaussian fields

A central limit theorem for the number of excursion set components of Gaussian fields

Limit theory for spatial random networks

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