Normal approximation in geometric probability

Penrose, Mathew D.; Yukich, J. E.

doi:10.1142/9789812567673_0003

Cited by 76 publications

(201 citation statements)

References 14 publications

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“…Adopt the convention D 1 (U 0 0 ) = 0. By the orthogonality of the Z j (Lemma 3.5) and (19), for 0 ≤ ℓ < m,…”

Section: Limit Behaviour For α =mentioning

confidence: 99%

“…Such a method was employed by Avram and Bertsimas [1] to give central limit theorems for nearest neighbour graphs and other random geometrical structures. A general version of this method is provided by [19]. By a similar argument to [1], one can show that, under * , the total weight (for α > 2/3) of edges in the MDST from points in the region (ε n , 1) 2 (for ε n given below) satisfies a central limit theorem, where…”

Section: Lemma 62 the Distribution Of ∆(∞) Is Non-degeneratementioning

confidence: 99%

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On the total length of the random minimal directed spanning tree

Penrose

Wade

2006

Adv. Appl. Probab.

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In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square, all edges must respect the "coordinatewise" partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.

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“…Adopt the convention D 1 (U 0 0 ) = 0. By the orthogonality of the Z j (Lemma 3.5) and (19), for 0 ≤ ℓ < m,…”

Section: Limit Behaviour For α =mentioning

confidence: 99%

Section: Lemma 62 the Distribution Of ∆(∞) Is Non-degeneratementioning

confidence: 99%

On the total length of the random minimal directed spanning tree

Penrose

Wade

2006

Adv. Appl. Probab.

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“…This enables central limit theorems formulated for just these situations, such as that of Baldi and Rinott (1989), to be applied. Penrose and Yukich (2005) combined their ideas with the general notion of a stabilizing functional and with the theorems of Chen and Shao (2004), obtaining very good rates of convergence for the central limit theorem in a wide range of problems of this kind. Their examples include the total edge length of the k-nearest-neighbour graph, the number of edges in the sphere-of-influence graph, and the independence number of the r-threshold graph, all based on the points of an underlying realization of a Poisson process in a bounded region of R d .…”

Section: Theorem 31 Under the Assumptions In The Preceding Paragrapmentioning

confidence: 99%

“…We begin by describing the setting of Penrose and Yukich (2005). We take H to be a marked Poisson process on = 1 × 2 , where 1 is a compact subset of R d and 2 is a mark space, assumed to be locally compact, second-countable, and Hausdorff.…”

Section: Local Dependence In Geometric Probabilitymentioning

confidence: 99%

Normal approximation for random sums

Barbour

Xia

2006

Adv. Appl. Probab.

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In this paper, we adapt the very effective Berry-Esseen theorems of Chen and Shao (2004), which apply to sums of locally dependent random variables, for use with randomly indexed sums. Our particular interest is in random variables resulting from integrating a random field with respect to a point process. We illustrate the use of our theorems in three examples: in a rather general model of the insurance collective; in problems in geometrical probability involving stabilizing functionals; and in counting the maximal points in a two-dimensional region.

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“…However, it does not seem to cover other functionals such as Vol K n or f j (K n ). For more information, see, e.g., Penrose and Yukich [47] and the references therein.…”

Section: This Follows Since For a Polytope Vol S(z T) Is Maximal Whmentioning

confidence: 99%

Random points and lattice points in convex bodies

Bárány

2008

Bull. Amer. Math. Soc.

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Abstract. Assume K ⊂ R d is a convex body and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes? How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases. The first is when X is a random sample of n uniform, independent points from K. In this case motivation comes from Sylvester's famous four-point problem and from the theory of random polytopes. The second case is when X = K ∩ Z d where Z d is the lattice of integer points in R d and the questions come from integer programming and geometry of numbers. Surprisingly (or not so surprisingly), the answers in the two cases are rather similar. Sylvester's four-point problemThe study of random points in convex bodies started with an innocent looking question. The year was 1864. The place was London. The journal was the Educational Times. Problem 1941 came from J. J. Sylvester [60]. It read: "Show that the chance of four points forming the apices of a reentrant quadrilateral is 1/4 if they be taken at random in an indefinite plane." Several answers came in. Most of them were different. In 1865 Sylvester [61] concluded, "This problem does not admit of a determinate solution." The reason is, as we all know by now, in "at random in an indefinite plane", since there is no natural probability measure on it. Sylvester immediately modified the question: Let K be a convex body in the plane and choose four random, independent, and uniform points from K. What is the probability that they form the vertices of a reentrant quadrilateral, or, in recent terminology, that their convex hull is a triangle. Further, for what K is this probability the smallest and the largest.This question has become known as Sylvester's four-point problem, and it has proved to be extremely fertile. It took more than fifty years (and several erroneous proofs) to find the answer. Blaschke showed in [26] and [27] that the probability in question is largest for a triangle and smallest for the disk (or any ellipse). His solution used the method of symmetrization and "shaking down" that have become standard tools since.Sylvester's question, and its subsequent solution, determined the direction of research for a long time. Many papers have been written starting with the setting: Let X n = {x 1 , ..., x n } be a random, independent, uniform sample of n points from

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Normal approximation in geometric probability

Cited by 76 publications

References 14 publications

On the total length of the random minimal directed spanning tree

On the total length of the random minimal directed spanning tree

Normal approximation for random sums

Random points and lattice points in convex bodies

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