Limit Theorems in Discrete Stochastic Geometry

Yukich, J. E.

doi:10.1007/978-3-642-33305-7_8

Cited by 20 publications

(21 citation statements)

References 59 publications

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“…In the stochastic geometry literature, there are several methods for obtaining a CLT; see e.g. Yukich (2013). The one we follow is the martingale method, which first writes network moments as a martingale difference sequence using the following spatial projection.…”

Section: H11 Poissonizationmentioning

confidence: 99%

Normal Approximation in Large Network Models

Leung

Moon

2019

SSRN Journal

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We prove central limit theorems for models of network formation and network processes with homophilous agents. The results hold under large-network asymptotics, enabling inference in the typical setting where the sample consists of a small set of large networks. We first establish a general central limit theorem under high-level "stabilization" conditions that provide a useful formulation of weak dependence, particularly in models with strategic interactions. The result delivers a ? n rate of convergence and a closed-form expression for the asymptotic variance. Then using techniques in branching process theory, we derive primitive conditions for stabilization in the following applications: static and dynamic models of strategic network formation, network regressions, and treatment effects with network spillovers. Finally, we suggest some practical methods for inference.JEL Codes: C31, C57, D85Network models have attracted considerable attention in economics as tractable representations of non-market interactions, such as peer effects and social learning, and formal economic relations, such as financial and trade networks. However, for many important network models, methods for inference are unavailable due to the lack of a large-sample theory. The novel feature of network data is that it typically takes the form of observations on a small set of large networks. Additionally, many network models of interest feature strategic interactions or spillovers, which generate dependence between network subunits. A large-sample theory must outline conditions under which the amount of "independent information" grows with the number of nodes or agents in the network, despite network autocorrelation.The main contribution of this paper is a large-network central limit theorem (CLT) applicable to a large class of network moments. These moments can be written as averages of node-level statistics, namelywhere X i is a vector of homophilous attributes for node i, 1 X n the set of all nodes' homophilous attributes, W the set of all other node attributes, and A the observed network or network time series on n nodes. A simple example is the degree of node i, ψpX i , X n , W, Aq " ř j‰i A ij , which is the number of links involving i. More generally, these moments may be complicated functionals of the network, including the average clustering coefficient, subnetwork counts, and regression estimators.There are two main technical contributions of the paper. First, we derive a new CLT that holds under high-level "stabilization" conditions on the node statistic ψpX i , X n , W, Aq. These provide a general formulation of network weak dependence for which we can derive primitive conditions in a variety of models. The main requirement is that i's node statistic only depends on a random "relevant set" of nodes with asymptotically bounded size that effectively constitutes i's dependency neighborhood. Our conditions are modifications of assumptions used in the stochastic geometry literature (e.g. Penrose and Yukich, 2001; Penrose, 2003), which f...

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Section: H11 Poissonizationmentioning

confidence: 99%

Normal Approximation in Large Network Models

Leung

Moon

2019

SSRN Journal

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“…We recall now the concept of a stabilizing functional which was introduced in [22-24] after earlier works [12,15]; see also the surveys [27,30]. Roughly speaking, a functional stabilizes if its value at a given point only depends on a local random neighbourhood and is unaffected by changes in point configurations outside of it.…”

Section: Admissible Score Functionsmentioning

confidence: 99%

“…Statistics of the Poisson-Voronoi approximation 13 against elements of C(∂A) (here, δ x stands for the unit-mass Dirac measure at x). The details of this extension are straightforward and may be found in, for example, [30], which deals with volume-order asymptotics for sums of score functions.…”

Section: Admissible Score Functionsmentioning

confidence: 99%

Asymptotic theory for statistics of the Poisson–Voronoi approximation

Thäle¹,

Yukich²

2016

Bernoulli

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This paper establishes expectation and variance asymptotics for statistics of the Poisson-Voronoi approximation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approximation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formula applicable in the setting of sums of stabilizing functionals.

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“…For example, TSP and MST (Minimum Spanning Tree) are smooth functionals. In [14], Yukich presents laws of large numbers for smooth, superadditive Euclidean functionals, see Theorem 8.1, as well as the general 'umbrella theorem' for subadditive, smooth Euclidean functionals, see Theorem 8.3.…”

Section: Introductionmentioning

confidence: 99%

“…Additionally, there is a large body of work, which uses the stabilization method, see Penrose [10], Yukich [14]. It is not hard to show that stabilization holds in the sub-critical regime for λ < λ c , when a.a.s 1 no infinite cluster exists, but it is not clear at all whether stabilization holds in the super-critical regime λ > λ c that we are interested in.…”

Section: Introductionmentioning

confidence: 99%

Scaling laws for maximum coloring of random geometric graphs

Borst

Bradonjic

2017

Discrete Applied Mathematics

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We examine maximum vertex coloring of random geometric graphs, in an arbitrary but fixed dimension, with a constant number of colors. Since this problem is neither scaleinvariant nor smooth, the usual methodology to obtain limit laws cannot be applied. We therefore leverage different concepts based on subadditivity to establish convergence laws for the maximum number of vertices that can be colored. For the constants that appear in these results, we provide the exact value in dimension one, and upper and lower bounds in higher dimensions.

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Limit Theorems in Discrete Stochastic Geometry

Cited by 20 publications

References 59 publications

Normal Approximation in Large Network Models

Normal Approximation in Large Network Models

Asymptotic theory for statistics of the Poisson–Voronoi approximation

Scaling laws for maximum coloring of random geometric graphs

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