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Nearest-neighbor graphs on the cantor set

Part of: Limit theorems Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Nathan Shank
Nathan Shank*
Affiliation:
Moravian College
*
∗ Postal address: Mathematics and Computer Science Department, 1200 Main Street, Bethlehem, PA 18018, USA. Email address: shank@math.moravian.edu
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Abstract

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Let be a collection of n uniform, independent, and identically distributed points on the Cantor ternary set. We consider the asymptotics for the expected total edge length of the directed and undirected nearest-neighbor graph on We prove convergence to a constant of the rescaled expected total edge length of this random graph. The rescaling factor is a function of the fractal dimension and has a log-periodic, nonconstant behavior.

Keywords

Nearest neighborCantor setlaw of large numbers

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 41 , Issue 1 , March 2009 , pp. 38 - 62
Copyright
Copyright © Applied Probability Trust 2009 

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