Gaussian polytopes: A cumulant-based approach

Grote, Julian; Thaele, Christoph

doi:10.1016/j.jco.2018.03.001

Cited by 18 publications

(15 citation statements)

References 38 publications

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“…It features in the proofs of moderate deviation principles and laws of the iterated logarithms for stabilizing functionals of Poisson point process [5], [22]. Fast decay of correlations (1.21) yields volume order cumulant bounds, useful in establishing concentration inequalities as well as moderate deviations, as explained in [27,Lemma 4.2]. (x) Normal approximation.…”

Section: Remarksmentioning

confidence: 99%

Limit theory for geometric statistics of point processes having fast decay of correlations

Błaszczyszyn

Yogeshwaran²,

Yukich³

2019

Ann. Probab.

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Let P be a simple, stationary point process on R d having fast decay of correlations, i.e., its correlation functions factorize up to an additive error decaying faster than any power of the separation distance. Let Pn := P ∩ Wn be its restriction to windows Wn := [− 1 2 n 1/d , 1 2 n 1/d ] d ⊂ R d . We consider the statistic H ξ n := x∈Pn ξ(x, Pn) where ξ(x, Pn) denotes a score function representing the interaction of x with respect to Pn. When ξ depends on local data in the sense that its radius of stabilization has an exponential tail, we establish expectation asymptotics, variance asymptotics, and central limit theorems for H ξ n and, more generally, for statistics of the re-scaled, possibly signed, ξ-weighted point measures µ ξ n := x∈Pn ξ(x, Pn)δ n −1/d x , as Wn ↑ R d . This gives the limit theory for non-linear geometric statistics (such as clique counts, the number of Morse critical points, intrinsic volumes of the Boolean model, and total edge length of the k-nearest neighbors graph) of α-determinantal point processes (for −1/α ∈ N) having fast decreasing kernels, including the β-Ginibre ensembles, extending the Gaussian fluctuation results of Soshnikov [72] to non-linear statistics. It also gives the limit theory for geometric U-statistics of α-permanental point processes (for 1/α ∈ N) as well as the zero set of Gaussian entire functions, extending the central limit theorems of Nazarov and Sodin [53] and Shirai and Takahashi [71], which are also confined to linear statistics. The proof of the central limit theorem relies on a factorial moment expansion originating in [12,13] to show the fast decay of the correlations of ξ-weighted point measures. The latter property is shown to imply a condition equivalent to Brillinger mixing and consequently yields the asymptotic normality of µ ξ n via an extension of the cumulant method.

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Section: Remarksmentioning

confidence: 99%

Limit theory for geometric statistics of point processes having fast decay of correlations

Błaszczyszyn

Yogeshwaran²,

Yukich³

2019

Ann. Probab.

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“…Gaussian sample is called the Gaussian polytope. The expected volume of the Gaussian polytope was computed by Efron ; see also and and for further recent results about Gaussian polytopes.…”

Section: Special Casesmentioning

confidence: 99%

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Kabluchko

Temesvari

Thäle

2018

Mathematische Nachrichten

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Let X1,⋯,Xn be i.i.d. random points in the d‐dimensional Euclidean space sampled according to one of the following probability densities: fd,βfalse(xfalse)=const·()1−∥x∥2β,false∥xfalse∥<1,(thebetacase)and truef∼d,βfalse(xfalse)=const·()1+∥x∥2−β,x∈double-struckRd,(thebeta"case).We compute exactly the expected intrinsic volumes and the expected number of facets of the convex hull of X1,⋯,Xn. Asymptotic formulae were obtained previously by Affentranger [The convex hull of random points with spherically symmetric distributions, 1991]. By studying the limits of the beta case when β↓−1, respectively β↑+∞, we can also cover the models in which X1,⋯,Xn are uniformly distributed on the unit sphere or normally distributed, respectively. We obtain similar results for the random polytopes defined as the convex hulls of ±X1,⋯,±Xn and 0,X1,⋯,Xn. One of the main tools used in the proofs is the Blaschke–Petkantschin formula.

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“…in the Gaussian case and Remark 3.6. Starting with the cumulant bounds presented in Theorem 3.1 one can also derive (i) concentration inequalities, (ii) bound for moments of all orders, (iii) Cramér-Petrov type results concerning the relative error in the central limit theorem, (iv) strong laws of large numbers for the random variables L n,r from the results presented in [33, Chapter 2] (see also [13,14]).…”

Section: 2mentioning

confidence: 99%

“…Denoting by ψ(t) the function defined at (14) and using (15) we conclude that, after simplification of the resulting terms, log Ee tLn,n = log ψ(t) + t m n + t 2 4 log…”

Section: 5mentioning

confidence: 99%

Limit theorems for random simplices in high dimensions

Grote¹,

Kabluchko²,

Thäle³

2019

ALEA

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Let r = r(n) be a sequence of integers such that r ≤ n and let X 1 , . . . , X r+1 be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on R n . Limit theorems for the log-volume and the volume of the random convex hull of X 1 , . . . , X r+1 are established in high dimensions, that is, as r and n tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-φ convergence are derived. The results heavily depend on the asymptotic growth of r relative to n. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if r = o(n) (respectively, r ∼ αn for some 0 < α < 1).2010 Mathematics Subject Classification. 52A22, 52A23, 60D05, 60F05, 60F10.

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Gaussian polytopes: A cumulant-based approach

Cited by 18 publications

References 38 publications

Limit theory for geometric statistics of point processes having fast decay of correlations

Limit theory for geometric statistics of point processes having fast decay of correlations

Expected intrinsic volumes and facet numbers of random beta‐polytopes

Limit theorems for random simplices in high dimensions

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