| Summary: | 碩士 === 國立中山大學 === 應用數學研究所 === 85 ===
Let F(x) be a probability distribution function. The Fourier transform
f(t)= ∫e dF(x), t E R,
of F(x) is called the "characteristic function of F(x)." It is known that characteristic function is an important tool in the probability theory and is veryuseful in the part of probability theory which can be studied independently of its measure-theoretic foundation.
In this thesis, we are interested in Stieltjes transforms of probability distribution functions. Let F(x) be a probability distribution function, its Stieltjes transform is defined by
SF(z) =∫ dF(x), where z=u+iv E C, v>0
We want to know if they have similar properties to the characteristic functions.
For the convenience of comparison, we list those properties of characteristic functions in Section 2, which are important in probability theory. In Section 3 and Section 4, we obtain some similar properties and theorems of Stieltjes transforms of probability distribution functions and give proofs.
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