Error bounds in stochastic-geometric normal approximation

Penrose, Mathew D.; Rosoman, Tom

doi:10.46298/dmtcs.3557

Cited by 3 publications

(6 citation statements)

References 20 publications

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“…Clearly (2.12) and (2.13) imply central limit theorems whereby both (V − µ V )/σ V and (S − µ S )/σ S converge in distribution to the standard normal, thereby providing an alternative to existing proofs of these central limit theorems [12,13,16]. In the Poissonized setting, nonasymptotic bounds analogous to those in Theorems 2.1 and 2.2 are given in [15] and imply O(n −1/2 ) bounds analogous to (2.12) and (2.13). In the de-Poissonized setting considered here, Chatterjee [3] provides bounds similar to those in (2.12) and (2.13), which hold for general metric spaces, but using the Kantorovich-Wasserstein distance, rather than the Kolmogorov distance considered here, and without providing any explicit constants.…”

Section: Resultsmentioning

confidence: 92%

“…For r > 0, we say that ψ has radius r if ψ(x, X ) is unaffected by the addition of points to, or removal of points from, the point set X at a distance more than r from x, that is, if for all (x, X ) we have ψ(x, X ) = ψ(x, X ∩ B r (x)). The notion of radius is the same as that of range of interaction used in [15]; see also the notion of radius of stabilization, in [15,17] and elsewhere. We also define…”

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confidence: 99%

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Normal approximation for coverage models over binomial point processes

Goldstein¹,

Penrose²

2010

Ann. Appl. Probab.

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We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over a binomial point process of $n$ points in a toroidal spatial region of volume $n$. The proof is based on Stein's method via size-biased couplings.Comment: Published in at http://dx.doi.org/10.1214/09-AAP634 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Resultsmentioning

confidence: 92%

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confidence: 99%

Normal approximation for coverage models over binomial point processes

Goldstein¹,

Penrose²

2010

Ann. Appl. Probab.

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“…(i) Comparing (2.17) with existing results. The results at (2.17) are applicable in the setting of volume order scaling of the variances, i.e., when the variances of H s and H ′ n exhibit scaling proportional to s and n. The rate for Poisson input in (2.17) significantly improves upon the rate given by Theorem 2.1 of [29] (see also Lemma 4.4 of [24]), Corollary 3.1 of [5], and Theorem 2.3 in [26], which all contain extraneous logarithmic factors and which rely on dependency graph methods. The rate in (2.17) for binomial input is new, as up to now there are no explicit general rates of normal convergence for sums of stabilizing score functions ξ n of binomial input.…”

Section: Resultsmentioning

confidence: 99%

“…In the papers [6,24,27] abstract central limit theorems for stabilizing functionals are derived and applied to several problems from stochastic geometry. Quantitative bounds for the normal approximation of stabilizing functionals of an underlying Poisson point process are given in [5,26,29,30,46]. These results yield rates of convergence for the Kolmogorov distance of the order 1/ √ Var H s times some extraneous logarithmic factors.…”

Section: Introductionmentioning

confidence: 99%

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Normal approximation for stabilizing functionals

Schulte¹,

Yukich²

2019

Ann. Appl. Probab.

147

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We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind.Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of R d , including m-dimensional Riemannian manifolds, m ≤ d. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the k-face and ith intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension d ≥ 2. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

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

Last

Peccati

Schulte

2015

Probab. Theory Relat. Fields

109

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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Error bounds in stochastic-geometric normal approximation

Cited by 3 publications

References 20 publications

Normal approximation for coverage models over binomial point processes

Normal approximation for coverage models over binomial point processes

Normal approximation for stabilizing functionals

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

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