Open AccessA law of large numbers for nearest neighbour statistics
Author(s) -
Dafydd Evans
Publication year - 2008
Publication title -
proceedings - royal society. mathematical, physical and engineering sciences
Language(s) - English
Resource type - Journals
eISSN - 1471-2946
pISSN - 1364-5021
DOI - 10.1098/rspa.2008.0235
Subject(s) - independent and identically distributed random variables , mathematics , law of large numbers , sample (material) , random variable , statistics , point (geometry) , sample mean and sample covariance , sample size determination , combinatorics , chemistry , geometry , chromatography , estimator
In practical data analysis, methods based on proximity (near-neighbour) relationships between sample points are important because these relations can be computed in time(n log n ) as the number of pointsn →∞. Associated with such methods are a class of random variables defined to be functions of a given point and its nearest neighbours in the sample. If the sample points are independent and identically distributed, the associated random variables will also be identically distributed but not independent. Despite this, we show that random variables of this type satisfy a strong law of large numbers, in the sense that their sample means converge to their expected values almost surely as the number of sample pointsn →∞.