Laws of large numbers and nearest neighbor distances

TL;DR

This paper investigates the behavior of power-weighted nearest neighbor distances in large samples from a multivariate density, providing criteria for their convergence and addressing inaccuracies in existing literature, with applications to entropy estimation.

Contribution

It offers new criteria ensuring law of large numbers for nearest neighbor distances, correcting previous inaccuracies and linking to entropy estimation.

Findings

Established conditions for convergence of nearest neighbor sums.

Corrected inaccuracies in prior literature.

Linked nearest neighbor distances to entropy estimation.

Abstract

We consider the sum of power weighted nearest neighbor distances in a sample of size n from a multivariate density f of possibly unbounded support. We give various criteria guaranteeing that this sum satisfies a law of large numbers for large n, correcting some inaccuracies in the literature on the way. Motivation comes partly from the problem of consistent estimation of certain entropies of f.