Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform

@article{Geronimo2003NecessaryAS,
title={Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform},
author={Jeffrey S. Geronimo and Theodore P. Hill},
journal={Journal of Approximation Theory},
year={2003},
volume={121},
pages={54-60}
}

The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel prob ability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) � −1 as y � ⊂, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.