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Necessary And Sufficient Condition That The Limit Of Stieltjes Transforms Is A Stieltjes Transform, Jeffrey S. Geronimo, Theodore P. Hill
Necessary And Sufficient Condition That The Limit Of Stieltjes Transforms Is A Stieltjes Transform, Jeffrey S. Geronimo, Theodore P. Hill
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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 iy S(iy) →−1as y →∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.