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

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• 作者：Geronimo ; Jeffrey S. ; Hill ; Theodore P.
• 关键词：Borel probability measure ; Weak convergence ; Stieltjes transform ; Grommer&ndash ; Hamburger theorem
• 刊名：Journal of Approximation Theory
• 出版年：2003
• 期刊代码：71_00219045
• 类别：mt
• 出版时间：March, 2003
• 卷：121
• 期：1
• 页码：54-60
• 文件大小：144 K

摘要

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