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. 

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