A Random Number Generator for the Kolmogorov Distribution

TL;DR

This paper presents an acceptance-rejection algorithm for generating random numbers from the Kolmogorov distribution, addressing convergence and efficiency issues related to its series-based density function.

Contribution

It introduces a novel acceptance-rejection method with proofs of convergence and boundedness for the Kolmogorov distribution's density function.

Findings

Proved uniform convergence of the series for the density function

Established bounds for the ratio between target and auxiliary densities

Proposed an optimal truncation method for the series expression

Abstract

We discuss an acceptance-rejection algorithm for the random number generation from the Kolmogorov distribution. Since the cumulative distribution function (CDF) is expressed as a series, in order to obtain the density function we need to prove that the series of the derivatives converges uniformly. We also provide a similar proof in order to show that the ratio between the target Kolmogorov density and the auxiliary density implemented is bounded. Finally we discuss a way of truncating the series expression of the density in an optimal way.