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

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