Computing Inverses of Stieltjes Transforms of Probability Measures

TL;DR

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 are established.

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.

Figures (11)

• Figure 1: The graph of $f$ (left) and its modified graph (right).
• Figure 2: Region where inverses can be recovered using $M_{(a,b)}(J(z))$ (Left) or $M_{(c,d)}(z)$ (Right), shaded in blue. As $r$ increases up to $1$, the union of the blue regions is the whole domain of $G_{\mu}$.
• Figure 3: Left: density of $\mu$ with Jacobi weight. Right: $4$ different $\zeta$ values and the curve $\gamma_1$.
• Figure 4: Error computed as $|G_{\mu}(\hat{z}_{i,j}) - \zeta_i|$ plotted against number of quadrature points $K$, for $r=0.9$ and $r=0.99$. The points $0.1+1.6\mathrm{i}$ and $-0.3-1.5\mathrm{i}$ both have $2$ inverses, whilst $-2+0.5\mathrm{i}$ and $0.2+0.4\mathrm{i}$ each only have $1$. The errors that grow in the left figure correspond to roots that lie outside the contour.
• Figure 5: Solutions of $G_{\mu}(z) = \zeta$ alongside the ellipses for $r=0.9$ and $r=0.99$.

Theorems & Definitions (47)

• Remark 1.0.1
• Lemma 2.1
• Theorem 2.2: Sokhotski--Plemelj theorem muskhelishvili1977singular
• Definition 2.1: Number of connected components
• Lemma 3.1
• proof
• Definition 3.1: Image of the support
• Lemma 3.2
• proof
• Proposition 3.3