Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform
The paper deals with the Stieltjes transform \(S_P\) of a probability measure \(P\) defined as the function \(S_P: \{z\in \mathbb{C}:\text{Im}(z)>0\}\to {\mathbb C}\) given by \[ S_P(z)=\int^{\infty}_{-\infty}{\frac{1}{w-z}}dP(w). \] The main result proved in Theorem 1 states that if \((P_n)\) are real Borel probability measures with Stieltjes transforms \((S_n)\), respectively, and if \(\lim_{n\to \infty}S_n(z)=S(z)\) for all \(z\) with \(\text{Im}(z)>0\), then there exists a Borel probability measure \(P\) with Stieltjes transform \(S_P=S\) if and only if \(\lim_{y\to \infty}[iyS(iy)]=-1\), in which case \(P_n\) converges to \(P\) in distribution. Examples are presented to Stieltjes transforms of discrete and atomic probability measures associated with orthogonal polynomials.
scientific article; zbMATH DE number 1915643
scientific article