The Speed of Convergence Under the Kolmogorov–Smirnov Metric in the Soshnikov Central Limit Theorem for the Sine Process

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 \(R>1\), 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.