設 F 為一機率分佈函數， F 的 Stieltjes 變換定義為 S_{F}(z)=int_{-infty}^{infty}frac{1}{t-z}dF(t)mbox{， 其中
}z=x+iyinmathbf{C}mbox{， }y>0mbox{。}

在這篇論文中，我們有興趣的是，什麼樣的 f 會是某一個 F 的
Stieltjes 變換。也就是說，我們想要知道當 f 滿足什麼條件時， f 可以寫成 int_{-infty}^{infty}frac{1}{t-z}dF(t)mbox{。}

Let F(t) be a probability distribution
function, its Stieltjes transform is defined by
S_{F}(z)=int_{-infty}^{infty}frac{1}{t-z}dF(t), where z=x+iyin$ {f C}, y>0.

In this thesis, we are interested in what f being the Stieltjes transform of some F. That is, we want to know what conditions f has, then f(z) can be written by int_{-infty}^{infty}frac{1}{t-z}dF(t).

1.Introduction
2.Stieltjes Transforms
3.Continuity Theorem of Stieltjes Transforms
4.Characterization of Stieltjes Transforms

1. H. S. Wall (1967). "Analytic Theory of Continued Fractions." Chelsea Pub. Company, New York.
2. Shern-Huh Ou (1997). "The Stieltjes Transforms of Probability Distribution Functions." Master thesis. National Sun Yat-sen University.
3. J. W. Silverstein Sang-Il Choi (1995). "Analysis of the limiting spectral distribution of large dimensional random matrices." Journal of Multivariate Analysis 54, 295-309.
4. T. Kawata (1972). ``Fourier Analysis in Probability Theory." Academic Press, New York London.
5. J. Weidmann (1980). "Linear Operators in Hilbert Spaces." Springer-Verlag, New York.