Closure of Random samples

TL;DR

This paper proves that the closure of a random sample in a k-dimensional space almost surely equals the set of all heavy points, which are points with neighborhoods of positive probability, revealing a deterministic structure in random samples.

Contribution

It establishes that the closure of a random sample converges to the set of heavy points, providing a new understanding of the geometric structure of random samples.

Findings

Closure of random samples equals the set of heavy points almost surely.

Heavy points are characterized by neighborhoods with positive probability.

The result links probabilistic properties with geometric structure.

Abstract

In this paper we show that the closure of a random sample for a k-dimensional random vector is almost surely a deterministic set of all heavy points of the distribution. A heavy point is defined to be a point for which all its neighborhoods have positive probability.