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The limit distribution of the maximum probability nearest-neighbour ball

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  30 July 2019

László Györfi ,
Norbert Henze  and
Harro Walk
László Györfi*
Affiliation:
Budapest University of Technology and Economics
Norbert Henze*
Affiliation:
Karlsruhe Institute of Technology (KIT)
Harro Walk*
Affiliation:
University of Stuttgart
*
*Postal address: Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Magyar Tudósok krt. 2., Budapest, H-1117, Hungary.
**Postal address: Institute of Stochastics, Karlsruhe Institute of Technology (KIT), Englerstr. 2, D-76133 Karlsruhe, Germany.
***Postal address: Institute of Stochastics and Applications, University of Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.
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Abstract

Let X1, …, Xn be independent random points drawn from an absolutely continuous probability measure with density f in ℝd. Under mild conditions on f, wederive a Poisson limit theorem for the number of large probability nearest-neighbour balls. Denoting by Pn the maximum probability measure of nearest-neighbour balls, this limit theorem implies a Gumbel extreme value distribution for nPn − ln n as n → ∞. Moreover, we derive a tight upper bound on the upper tail of the distribution of nPn − ln n, which does not depend on f.

Keywords

Nearest neighbourGumbel extreme value distributionPoisson limit theoremexchangeable event

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G09: Exchangeability 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 574 - 589
Copyright
© Applied Probability Trust 2019 

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