Balancing fractional Brownian motion

2026-08-10

AI Summary

This work investigates the discrepancy minimization problem for balancing $n$ independent sample paths of fractional Brownian motion over the interval $[0,1]$, serving as an infinite-dimensional generalization of Gaussian vector balancing. Leveraging wavelet representations of fractional Brownian motion and truncated balancing analysis, combined with probabilistic methods and randomized optimization algorithms, the study uncovers a phase transition at the Hurst index $H = 1/2$. Specifically, it establishes discrepancy bounds of $O(n^{1/2 - H} \sqrt{\log n})$, $O((\log n)^{3/2})$, and $O(\sqrt{\log n})$ for $H < 1/2$, $H = 1/2$, and $H > 1/2$, respectively, and proves an information-theoretic lower bound of $\Omega(n^{1/2 - H})$. The geometry of the solution space in the critical case is also characterized—such as the number of local minima and overlap gaps—and a polynomial-time algorithm is proposed that applies across the entire parameter range.

Abstract

We study the discrepancy of balancing $n$ independent sample paths of fractional Brownian motion with Hurst exponent $H\in(0,1)$ on $[0,1]$, an infinite-dimensional analogue of balancing Gaussian vectors. We establish a phase transition at $H=1/2$: with high probability, the discrepancy is $Ω(n^{1/2-H})$ and $\mathcal O(n^{1/2-H}(\log n)^{c(H)})$, where $c(H)=H+1/2$ if $H\geq 1/2$ and $c(H)=1/2$ otherwise. At the critical exponent $H=1/2$, we show that the discrepancy is $Θ(1)$ with constant probability as $n\to\infty$. In this regime, we further characterize the geometry of the solution space by computing the expected number of local minima, establishing an overlap gap property near the existence threshold, and proving its absence at every diverging optimality threshold. We also give randomized polynomial-time algorithms that compute signings with discrepancy $\mathcal O(n^{1/2-H}\sqrt{\log n})$ for $H<1/2$, $\mathcal O((\log n)^{3/2})$ for $H=1/2$, and $\mathcal O(\sqrt{\log n})$ for $H>1/2$, with high probability. Our analysis combines a truncated balancing argument based on a wavelet representation of fractional Brownian motion with probabilistic methods.

Problem

• fractional Brownian motion
• discrepancy
• balancing
• Hurst exponent
• phase transition

Innovation

• fractional Brownian motion
• discrepancy minimization
• phase transition
• overlap gap property
• randomized algorithms