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Asymptotic theory for the multidimensional random on-line nearest-neighbour graph

Wade, Andrew R.

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Abstract

The on-line nearest-neighbour graph on a sequence of n uniform random points in (0,1)d (d∈N) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent α∈(0,d/2], we prove O(max{n1−(2α/d),logn}) upper bounds on the variance. On the other hand, we give an n→∞ large-sample convergence result for the total power-weighted edge-length when α>d/2. We prove corresponding results when the underlying point set is a Poisson process of intensity n.

Citation

Wade, A. R. (2009). Asymptotic theory for the multidimensional random on-line nearest-neighbour graph. Stochastic Processes and their Applications, 119(6), 1889-1911. https://doi.org/10.1016/j.spa.2008.09.006

Journal Article Type Article
Publication Date Jun 1, 2009
Deposit Date Oct 4, 2012
Publicly Available Date Jan 31, 2013
Journal Stochastic Processes and their Applications
Print ISSN 0304-4149
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 119
Issue 6
Pages 1889-1911
DOI https://doi.org/10.1016/j.spa.2008.09.006
Keywords Random spatial graphs, Network evolution, Variance asymptotics, Martingale differences.

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Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Stochastic processes and their applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Stochastic processes and their applications, 119(6), 2009, 10.1016/j.spa.2008.09.006




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