Asymptotic normality of the number of small distances between random points in a cube

Kester, Adri

doi:10.1016/0304-4149(75)90005-8

Cited by 15 publications

(5 citation statements)

References 3 publications

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“…It would seem that a random graph approach provides a potentially useful and unifying way to model discrete spatial processes. This is evident to some extent in the literature; see, for instance, Zhou and Jammalamadaka [18], Jammalamadaka and Janson [11], Silverman and Brown [16], Kester [12], and various references therein. The main focus seems to have been on asymptotic distribution theory for the number of edges in the random graph.…”

Section: Introductionmentioning

confidence: 85%

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d

Appel

Russo

1997

Advances in Applied Probability

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On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.

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Section: Introductionmentioning

confidence: 85%

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d

Appel

Russo

1997

Advances in Applied Probability

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“…By [21], the probability that a point is within r of u is approximately 1 − e −αr m for some α. If the sample U is large enough, then the set of nearest neighbour distances will be sufficiently independent [14,18] that their empirical cumulative distribution will also be approximately equal to 1 − e −αr m . If D is the ordered sequence of nearest neighbour distances, then L D * (r) in Definition 3 will be approximately 1 − e −βr m for some β and 1 − L D * (r) will be approximately e −βr m…”

Section: Simulated Values Of ρ 2 (U ) For Various Random Variablesmentioning

confidence: 99%

Warp statistics and financial returns

Ralph

2016

Journal of Statistical Computation and Simulation

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We present a new approach to financial returns based on an infinite family of statistics called slide statistics that we introduce. The evidence these statistics provide suggests that certain distributions such as the stable distributions are not good models for the financial returns from various securities or indexes like the S&P 500 and the Dow Jones. Formally, we associate with any finite subset of a metric space an infinite sequence of scale invariant numbers ρ 1 , ρ 2 , . . . derived from a variant of differential entropy called the genial entropy. We give explicit formulas for ρ 1 and ρ 2 that are easily evaluated by a computer and make this theory particularly suitable for applications. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where 1/ρ 1 converges to the Hausdorff dimension and we prove that ρ 1 ≥ 0. For a uniform random variable X on [a, b] n , the evidence from simulations suggests that ρ 1 (X) = 1/n and ρ 2 (X) = −π 2 /(6n 2 ) which yields new tests for spatial randomness. The slide statistics describe continuous random variables in an entirely new way. For example, if Z is any normal variable then simulations suggest that ρ 1 (Z) = 4/π and ρ 2 (Z) = −1 which provides new goodness of fit tests for normality.

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“…By [15], the probability that a point is within r of x is approximately 1 − e −αr m for some α. If the sample U is large enough, then the set of nearest neighbour distances will be sufficiently independent [9,13] that their empirical cumulative distribution will also be approximately equal to 1−e −αr m . If D is the ordered sequence of nearest neighbour distances, then L D * (r) in Definition 3 will be approximately 1 − e −βr m for some β and 1 − L D * (r) will be approximately e −βr m .…”

Section: Simulated Values Of ρ 2 (U ) For Various Random Variablesmentioning

confidence: 99%

The Slide Dimension of Point Processes

Ralph

2014

Preprint

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We associate with any finite subset of a metric space an infinite sequence of scale invariant numbers ρ 1 , ρ 2 , . . . derived from a variant of differential entropy called the genial entropy. As statistics for point processes, these numbers often appear to converge in simulations and we give examples where 1/ρ 1 converges to the Hausdorff dimension. We use the ρn to define a new notion of dimension called the slide dimension for a special class of point processes on metric spaces. The slide calculus is developed to define ρn and an explicit formula is derived for the calculation of ρ 1 . For a uniform random variable X on [0, 1] n , evidence is given that ρ 1 (X) = 1/n and ρ 2 (X) = −π 2 /(6n 2 ) and simulations with a normal variable Z suggest that ρ 1 (Z) = 4/π and ρ 2 (Z) = −1. Some potential applications to spatial statistics are considered.

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Asymptotic normality of the number of small distances between random points in a cube

Cited by 15 publications

References 3 publications

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d

Warp statistics and financial returns

The Slide Dimension of Point Processes

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Asymptotic normality of the number of small distances between random points in a cube

Cited by 15 publications

References 3 publications

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]d

Warp statistics and financial returns

The Slide Dimension of Point Processes

Contact Info

Product

Resources

About

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d

The Maximum Vertex Degree of a Graph on Uniform Points in [0, 1]^d