A Lower Bound for the Discrepancy of a Random Point Set

TL;DR

This paper establishes a probabilistic lower bound on the discrepancy of random point sets in high-dimensional cubes, showing that they typically deviate from uniformity by a factor related to the square root of dimension and number of points.

Contribution

It provides a new lower bound for the star discrepancy of random point sets, revealing inherent limitations in their uniformity in high dimensions.

Findings

Expected star discrepancy is of order rac{s}{N}

With high probability, random sets contain axis-aligned rectangles with excess points proportional to rac{sN}

Discrepancy bounds hold with probability at least 1 - rac{(s)}

Abstract

We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s \le N$, the point set consisting of $N$ points chosen uniformly at random in the $s$-dimensional unit cube $[0,1]^s$ with probability at least $1-\exp(-\Theta(s))$ admits an axis parallel rectangle $[0,x] \subseteq [0,1]^s$ containing $K \sqrt{sN}$ points more than expected. Consequently, the expected star discrepancy of a random point set is of order $\sqrt{s/N}$.