# 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.

## Key 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.