Computing Inverses of Stieltjes Transforms of Probability Measures

TL;DR

This paper develops conditions and algorithms for rigorously computing all inverses of the Stieltjes transform of probability measures, which is crucial in applications like free probability.

Contribution

It introduces bounds on the number of inverses based on measure properties and combines these with contour integral algorithms for accurate computation.

Findings

Established bounds on the number of inverses of the Stieltjes transform.

Developed contour integral-based algorithms for computing all inverses.

Applicable to problems in free probability and related fields.

Abstract

The Stieltjes (or sometimes called the Cauchy) transform is a fundamental object associated with probability measures, corresponding to the generating function of the moments. In certain applications such as free probability it is essential to compute the inverses of the Stieltjes transform, which might be multivalued. This paper establishes conditions bounding the number of inverses based on properties of the measure which can be combined with contour integral-based root finding algorithms to rigorously compute all inverses.