The central limit theorem for Euclidean minimal spanning trees II

Lee, Sungchul

doi:10.1239/aap/1029955253

Cited by 28 publications

(14 citation statements)

References 14 publications

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“…We use the notation of Section 2.2: A is a σ(X )-measurable random field on R d , where X is a completely independent random field on some measure space X = x∈Z d ,t∈Z l X x,t with values in some measurable space M . The following definition is inspired by the notion of stabilization radius first introduced by Lee [27,28] and crucially used in the works by Penrose, Schreiber, and Yukich on random sequential adsorption processes [35,34,36,37]. Definition 2.5.…”

Section: 2mentioning

confidence: 99%

Multiscale functional inequalities in probability: Concentration properties

Duerinckx¹,

Gloria²

2020

ALEA

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Consider an ergodic stationary random field A on the ambient space R d . In order to establish concentration properties for nonlinear functions Z(A), it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.

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Section: 2mentioning

confidence: 99%

Multiscale functional inequalities in probability: Concentration properties

Duerinckx¹,

Gloria²

2020

ALEA

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“…As made clear by the title and by the previous discussion, we regard the hypothesis (1.8) of Proposition 1.3 as a weak form of stabilization. The powerful and farreaching concept of stabilization in the context of central limit theorems was introduced in its actual form by Penrose and Yukich in [42,43] and Baryshnikov and Yukich in [5], building on the set of techniques introduced by Kesten and Lee [22] and Lee [29]. This notion typically applies to a collection of geometric functionals {F t : t ≥ 1} of the type…”

Section: Further Connections With the Existing Literaturementioning

confidence: 99%

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization

Last

Peccati

Schulte

2015

Probab. Theory Relat. Fields

107

234

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We prove a new class of inequalities, yielding bounds for the normal approximation in the Wasserstein and the Kolmogorov distance of functionals of a general Poisson process (Poisson random measure). Our approach is based on an iteration of the classical Poincaré inequality, as well as on the use of Malliavin operators, of Stein's method, and of an (integrated) Mehler's formula, providing a representation of the Ornstein-Uhlenbeck semigroup in terms of thinned Poisson processes. Our estimates only involve first and second order difference operators, and have consequently a clear geometric interpretation. In particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization, and with non-linear functionals of Poisson shot-noise processes. We discuss two examples of stabilizing functionals in great detail: (i) the edge length of the k-nearest neighbour graph, (ii) intrinsic volumes of k-faces of Voronoi tessellations. In all these examples we obtain rates of convergence (in the Kolmogorov and the Wasserstein distance) that one can reasonably conjecture to be

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“…More generally, we will prove a limit theorem for a large class of functionals, called stabilizing functionals. This class was first introduced by [18] and it was used by Penrose and Yukich [23,24]; it is slightly modified here to suit to our framework. Roughly speaking, F (X, T ) stabilizes T if the value at X depends only a small number of vertices around X.…”

mentioning

confidence: 99%

The radial spanning tree of a Poisson point process

Baccelli¹,

Bordenave²

2007

Ann. Appl. Probab.

161

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We analyze a class of spatial random spanning trees built on a realization of a homogeneous Poisson point process of the plane. This tree has a simple radial structure with the origin as its root. We first use stochastic geometry arguments to analyze local functionals of the random tree such as the distribution of the length of the edges or the mean degree of the vertices. Far away from the origin, these local properties are shown to be close to those of a variant of the directed spanning tree introduced by Bhatt and Roy. We then use the theory of continuous state space Markov chains to analyze some nonlocal properties of the tree, such as the shape and structure of its semi-infinite paths or the shape of the set of its vertices less than $k$ generations away from the origin. This class of spanning trees has applications in many fields and, in particular, in communications.Comment: Published at http://dx.doi.org/10.1214/105051606000000826 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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The central limit theorem for Euclidean minimal spanning trees II

Cited by 28 publications

References 14 publications

Multiscale functional inequalities in probability: Concentration properties

Multiscale functional inequalities in probability: Concentration properties

Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization

The radial spanning tree of a Poisson point process

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