The diameter of an elliptical cloud

Demichel, Yann; Fermin, Ana Karina; Soulier, Philippe

doi:10.1214/ejp.v20-3777

Cited by 4 publications

(2 citation statements)

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“…In the unbounded case, Henze and Lao [6] obtained a (non-Gumbel) limit distribution of M n if the distribution of Z is power-tailed spherically decomposable. Finally, Demichel et al [4] proved several results for the diameter of an elliptical cloud with unbounded support.…”

Section: Introductionmentioning

confidence: 92%

The limit distribution of the largest interpoint distance for distributions supported by ad-dimensional ellipsoid and generalizations

Schrempp

2016

Adv. Appl. Probab.

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We study the asymptotic behaviour of the maximum interpoint distance of random points in a planar bounded set with an unique major axis and a boundary behaving like an ellipse at the endpoints. Our main result covers the case of uniformly distributed points in an ellipse.Keywords: Maximum interpoint distance, geometric extreme value theory, Poisson process, uniform distribution in an ellipse IntroductionFor some fixed integer d ≥ 2, let X 1 , X 2 , . . . be a sequence of independent and identically distributed (i.i.d.) d-dimensional random vectors, defined on a common probability space (Ω, A, P). Writing | · | for the Euclidean norm on R d , the convergence in distribution of the suitably normalized maximum interpoint distancehas been a topic of interest for more than 20 years. Results obtained so far are mostly for the case that the distribution P X1 of X 1 is spherically symmetric, and they may roughly be classified according to whether P X1 has an unbounded or a bounded support. If X 1 has a spherically symmetric normal distribution, Matthews and Rukhin (1993) obtained a Gumbel limit distribution for M n . Henze and Klein (1996) generalized this result to the case that X 1 has a spherically symmetric Kotz distribution. An even more general spherically symmetric setting has recently been studied by Jammalamadaka and Janson (2015). In the unbounded case, Henze and Lao (2015) obtained a (non-Gumbel) limit distribution of M n if the distribution of X 1 is power-tailed spherically decomposable. This case covers certain long-tailed spherically symmetric distributions for X 1 . Finally, Demichel et al. (2014) proved several results for the diameter of an elliptical cloud.If P X1 has a bounded support, Appel et al. (2002) obtained a convolution of two Weibull distributions as limit law of M n if X 1 has uniform distribution in a planar set with unique major axis and sub-√ x decay of its boundary at the endpoints. Moreover, they derived bounds for the limit law of M n if X 1 has a uniform distribution in an ellipse. Lao (2010), and Mayer and Molchanov (2007) obtained Weibull limit distributions for M n under very general settings if the distribution of X 1 is supported by the d-dimensional unit ball B d for d ≥ 2 (including the case of a uniform distribution). Lao (2010) obtained limit laws for M n if P X1 is uniform or non-uniform in the unit square, uniform in regular polygons, or uniform in the unit d-cube, d ≥ 2. Moreover, if P X1 is uniform in a proper ellipse, she improved the lower bound on the limit distribution of M n given in Appel et al. (2002). The limit behaviour of M n if P X1 is uniform in a proper ellipse has been an open problem for many years. Without giving a proof, Jammalamadaka and Janson (2015) state that n 2/3 (2 − M n ) has a limit distribution (involving two independent Poisson processes) if X 1 has a uniform distribution in a proper ellipse with major axis 2. We generalize this result to the case that the distribution is uniform or non-uniform over a planar bounded set satisfying certain regul...

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Section: Introductionmentioning

confidence: 92%

The limit distribution of the largest interpoint distance for distributions supported by ad-dimensional ellipsoid and generalizations

Schrempp

2016

Adv. Appl. Probab.

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“…For the unbounded support case, Henze (2010) showed that the limiting distribution of M n is not of the Gumble type if the distribution of X is power-tailed spherically decomposable. Demichel et al (2015) consider the elliptical distributions with unbounded support. In the case of bounded support, Tang et al (2022) provide a comprehensive review of the current body of literature.…”

Section: Introductionmentioning

confidence: 99%