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Mutual and shared neighbor probabilities: finite- and infinite-dimensional results

Published online by Cambridge University Press:  01 July 2016

M. F. Schilling
M. F. Schilling*
Affiliation:
California State University, Northridge
*
∗Postal address: Department of Mathematics, School of Science and Mathematics, California State University, Northridge, 18111 Nordhoff St, Northridge, CA 91330, USA.
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Abstract

Let X1, ···, Xn be i.i.d. random variables defined in ℝd having common continuous density f(x), and let Rij be the rank of Xj in the ordered list of distances from X¡. Both the mutual neighbor probabilities p1(r, s) = P(R12 = r, R21 = s) and the neighbor-sharing probabilities p2(r, s) = P(R13 = r, R23 = s) are studied from an asymptotic viewpoint. Infinite-dimensional limits are found for both situations and take particularly simple forms. Both cases exhibit considerable stability across dimensions and thus are well approximated by their infinite-dimensional values. Tables are provided to support the results given.

Keywords

NEAREST NEIGHBORSGEOMETRIC PROBABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 388 - 405
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research supported in part by National Science Foundation Grants MCS79-19141, MCS80-17103.

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