Random polytopes: Central limit theorems for intrinsic volumes

Thäle, Christoph; Turchi, Nicola; Wespi, Florian

doi:10.1090/proc/14000

Cited by 24 publications

(25 citation statements)

References 21 publications

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“…Examples include the expected intrinsic volumes or the expected face numbers. For these functionals also variance asymptotics and accompanying central limit theorems are available; see [7,8,27,34]. We refer the reader to the survey articles [5,14,28] and the book of Schneider and Weil [33] for more background material, information and references.…”

Section: Introduction and Description Of The Main Results 1introductionmentioning

confidence: 99%

A new approach to weak convergence of random cones and polytopes

Kabluchko¹,

Temesvari²,

Thäle

2020

Can. J. Math.-J. Can. Math.

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A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation generated by n independent and uniformly distributed random linear hyperplanes in R d+1 weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in R d , as n → ∞.

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Section: Introduction and Description Of The Main Results 1introductionmentioning

confidence: 99%

A new approach to weak convergence of random cones and polytopes

Kabluchko¹,

Temesvari²,

Thäle

2020

Can. J. Math.-J. Can. Math.

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“…. , d}, have been studied, for example, in [14], variance bounds can be found in [2] and [4], and central limit theorems were treated in [11], [15], [22], and [24]. More details can be found in the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In this last section we prove the central limit theorem. In contrast to [22], where floating bodies were used, here we work with surface bodies, as was already done in [21] for the case of the volume. In addition, we make use of the normal approximation bound of Proposition 2.…”

Section: Central Limit Theoremmentioning

confidence: 99%

Limit theorems for random polytopes with vertices on convex surfaces

Turchi

Wespi

2018

Adv. Appl. Probab.

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The random polytope K n , defined as the convex hull of n points chosen uniformly at random on the boundary of a smooth convex body, is considered. Proofs for lower and upper variance bounds, strong laws of large numbers and central limit theorems for the intrinsic volumes of K n are presented. A normal approximation bound from Stein's method and estimates for surface bodies are among the involved tools.Intrinsic volumes have been studied extensively in the alternative setting of random polytopes that arise as convex hulls of points chosen uniformly at random inside a fixed convex body. Results concerning the expectation of V ℓ (K n ), ℓ ∈ {1, . . . , d}, have been studied, for example, by Reitzner [10], variance bounds can be found in Böröczky, Fodor, Reitzner and Vígh [4] and Bárány, Fodor and Vígh [1], and central limit theorems were treated in Reitzner [11], Vu [17], Lachièze-Rey, Schulte and Yukich [7] and Thäle, Turchi and Wespi [16]. More details can be found in the references therein.On the other hand, the approximation of a convex body K, by means of a sequence of random polytopes K n , is improved whenever the vertices of K n are restricted to lie on the boundary of K, therefore making it a model worth studying. Indeed, in this framework the expectations ofd}, have been studied, for example, by Buchta, Müller and Tichy [5], Reitzner [8], Schütt and Werner [14] and Böröczky, Fodor and Hug [3]. However, more detailed informations are only known about the distribution of the volume V d (K n ). In particular, an upper variance bound was found by Reitzner [9] and a lower variance bound together with concentration inequalities by Richardson, Vu and Wu [12]. Only recently, Thäle [15] obtained a quantitative central limit theorem for V d (K n ) based on Stein's method.Our first aim is to generalize the results obtained in [9, 12] to V ℓ (K n ), ℓ ∈ {1, . . . , d}. In fact, we prove lower variance bounds following the ideas of [1,11,12] and upper variance bounds in the manner of [1], making use of the Efron-Stein jackknife inequality from [9]. In particular, the upper variance bounds imply strong laws of large numbers as in [1]. Secondly, we prove quantitative central limit theorems for V ℓ (K n ), ℓ ∈ {1, . . . , d}, using a normal approximation bound obtained in [6], extending the result of [15]. We now introduce some notation in order to present our results. Let (a n ) n∈N and (b n ) n∈N be two sequences of real numbers. We write a n ≪ b n (or a n ≫ b n ) if there exist a constant c ∈ (0, ∞) and a positive number n 0 such that a n ≤ c b n (or a n ≥ c b n ) for all n ≥ n 0 . Furthermore, a n = Θ(b n ) means that b n ≪ a n ≪ b n .Our first result concerns asymptotic lower and upper bounds, respectively, for the variances of the intrinsic volumes.Theorem 1.1. Let K ∈ K 2 + and choose n random points on ∂K independently and according to the probability distribution H d−1 . Then,Based on a result stated in [8, Theorem 1] concerning the behaviour of V ℓ (K) − E[V ℓ (K n )], the upper variance bounds of Theorem 1.1 ...

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“…The crucial part in the proof of Theorem 1.1 is the application of the general bound given by Theorem 2.2, thus we need to investigate the moments occuring in τ 1 , τ 2 and τ 3 . In the first step, we adapt and slightly extend the proof from [TTW18] for the binomial case, to work in the Poisson case, yielding upper bounds on the moments of the first and second order difference operators applied to the intrinsic volumes V j (K t ) which will be used in the second step to derive the bounds for the valuation functional. First order difference operator: Fix x ∈ K and j ∈ {1, .…”

Section: Proofmentioning

confidence: 99%

“…We start to investigate the moments of the first and second order difference operators applied to the components of the f -vector by combining combinatorial results from [Rei05b] with the floating body and economic cap covering approach from [Rei10] and [TTW18].…”

Section: Proof Of Theorem 14: Multivariate Functionalmentioning

confidence: 99%

Multivariate Normal Approximation for Functionals of Random Polytopes

Grygierek

2020

J Theor Probab

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Consider the random polytope, that is given by the convex hull of a Poisson point process on a smooth convex body in R d . We prove central limit theorems for continuous motion invariant valuations including the Will's functional and the intrinsic volumes of this random polytope. Additionally we derive a central limit theorem for the oracle estimator, that is an unbiased an minimal variance estimator for the volume of a convex set. Finally we obtain a multivariate limit theorem for the intrinsic volumes and the components of the f -vector of the random polytope.

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Random polytopes: Central limit theorems for intrinsic volumes

Abstract: Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability theory.

Cited by 24 publications

References 21 publications

A new approach to weak convergence of random cones and polytopes

A new approach to weak convergence of random cones and polytopes

Limit theorems for random polytopes with vertices on convex surfaces

Multivariate Normal Approximation for Functionals of Random Polytopes

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