Year: 2025
Type: article
Abstract: It is well understood that if one is given a set $$X \subset [0,1]$$ X ⊂ [ 0 , 1 ] of n independent uniformly distributed random variables, then $$\begin{aligned} \sup _{0 \le x \le 1} \left| \frac{\# X \cap [0,x]}{\# X} - x \right| \lesssim \frac{\sqrt{\log {n}}}{ \sqrt{n}} \qquad \text{ with } \text{ high } \text{ probability. } \end{aligned}$$ sup 0 ≤ x ≤ 1 # X ∩ [ 0 , x ] # X - x ≲ log n n with high probability. We show that one can improve the error term by removing a few of the points. For any $$m \le 0.001n$$ m ≤ 0.001 n there exists a subset $$Y \subset X$$ Y ⊂ X obtained by deleting at most m points, so that the error term drops from $$\sim \sqrt{\log {n}}/\sqrt{n}$$ ∼ log n / n to $$ \log {(n)}/m$$ log ( n ) / m with high probability. When $$m=cn$$ m = c n for a small $$0 \le c \le 0.001$$ 0 ≤ c ≤ 0.001 , this achieves the essentially optimal asymptotic order of discrepancy $$\log (n)/n$$ log ( n ) / n . The proof is constructive and works in an online setting (where one is given the points sequentially, one at a time, and has to decide whether to keep or discard it). A change of variables shows the same result for any random variables on the real line with absolutely continuous density.
Source: Journal of Geometric Analysis
Authors: Dmitriy Bilyk, Stefan Steinerberger
Institutions: University of Minnesota, University of Washington
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,290
OpenAlex ID: W4414225482