The Speed of Convergence with respect to the Kolmogorov-Smirnov Metric in the Soshnikov Central Limit Theorem for the Sine-Process

TL;DR

This paper investigates the rate at which rescaled additive functionals of the sine-process converge to a Gaussian distribution, providing explicit bounds on the Kolmogorov-Smirnov distance depending on the function's properties.

Contribution

It establishes new upper bounds on the convergence speed in the Kolmogorov-Smirnov metric for the sine-process, depending on the smoothness or holomorphicity of the test functions.

Findings

Bound of c/ log R for smooth functions

Bound of c/R for holomorphic functions

Convergence rate depends on function regularity

Abstract

For rescaled additive functionals of the sine-process, upper bounds are obtained for their speed of convergence to the Gaussian distribution with respect to the Kolmogorov-Smirnov metric. Under scaling with coefficient $R$ the Kolmogorov-Smirnov distance is bounded from above by $c/\log R$ for a smooth function and by $c/R$ for a function holomorphic in a horizontal strip.