Computing Inverses of Stieltjes Transforms of Probability Measures

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.

• The paper’s main contribution is establishing upper bounds on the number of inverses a Stieltjes transform can have using contour integration methods.
• It employs rigorous analytic techniques, including contour integration and Hilbert transform analysis, to address multivalued inverses in complex measures.
• These methods provide actionable insights for applications in free probability and random matrix theory, particularly in computing eigenvalue distributions.

Computing Inverses of Stieltjes Transforms of Probability Measures: An Analytical Approach

This paper explores a rigorous analysis of the computation of inverses of Stieltjes transforms associated with probability measures. Authored by James Chen and Sheehan Olver, the study addresses a critical requirement in applications such as free probability, where evaluating such inverses is prevalent. The Stieltjes transform typically corresponds to the generating function of the moments of a measure and is foundational in various analytical domains, including spectral theory and orthogonal polynomials.

Key Contributions and Results

The primary contribution of this work is the establishment of conditions that provide upper bounds on the number of inverses that a Stieltjes transform can possess. This result is significant, particularly in scenarios where the inverses might be multivalued. The authors utilize contour integral-based root-finding algorithms, offering a method to compute all inverses rigorously. The theoretical framework is supported by bounding arguments in the complex plane, which connect the structure of the measure with the analytic properties of its Stieltjes transform.

Analytical Framework

1. Stieltjes Transform: Given a Borel probability measure $\mu$ on $\mathbb{R}$, its Stieltjes transform $G_{\mu}(z)$ is defined for $z \in \mathbb{C} \setminus \Gamma$, where $\Gamma$ is the support of $\mu$.
2. Hilbert Transform: The paper also defines the Hilbert transform, relating the transform to the density and Stieltjes transform analytically.
3. Conditions for Multivalued Inverses: The study introduces criteria based on the support and structure of the measure to determine the number of possible inverses. Rigorous bounds derived therein are particularly useful for measures with disconnected support or with jump discontinuities.

Computational Methodology

The authors describe a contour-integration approach for computing the inverses of the Stieltjes transforms. They emphasize the efficacy of these methods for measures that are supported on multiple intervals or have certain discontinuities in density, offering a remedy to challenges arising from non-univalent situations.

Implications and Applications

The computational methods and theoretical insights provided are vital for applications in free probability theory. In these domains, the Stieltjes and related transforms, such as the $R$-transform, enable the characterization of free convolutions of measures. Specifically, the practical implications are noteworthy in the context of computing eigenvalue distributions of large Hermitian random matrices, important in fields like statistical physics and random matrix theory.

Future Prospects

The paper opens avenues for further research in numerically robust and efficient computations of free convolutions for more complex measures. The techniques developed could be extended or refined to accommodate an even broader class of probability measures, including those with more intricate support structures or mix of absolutely continuous and discrete components.

In conclusion, Chen and Olver's investigation provides a comprehensive analytical and computational framework for tackling the complexities associated with the inverses of Stieltjes transforms, with significant implications for both theoretical advancements and computational practices in applied mathematics.