A law of large numbers for nearest neighbour statistics

Abstract

In practical data analysis, methods based on proximity (near-neighbour) relationships between sample points are important because these relations can be computed in time O(n log n) as the number of points n → ∞. 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 points n → ∞. © 2008 The Royal Society.