特征函数与分布函数之比较评价

Comparison Evaluation between Characteristic Function and Distribution Function

本文从归纳总结分布函数和特征函数的重要作用入手,着重强调了它们之间有着重要的联系,通常我们认为分布函数可以决定特征函数,同时由唯一性定理知道特征函数也可以决定分布函数。但通过本文引入的反例,我们可以知道,唯一性定理在有限区间上不成立,不过在无穷限区间上我们仍可以说,分布函数和特征函数是一一对应的。

This paper starts from concluding and summarizing the important function of distribution function and characteristic function, emphasizing they have important contact. It is a usual presumption that distribution function can decide characteristic function and characteristic function can also decide the distribution function by the uniqueness theorem. But through the counter illustration in this article, it is demonstrated that the uniqueness theorem doesnt apply in the limited interval, however, it still can be concluded that distribution function and characteristic function are one-to-one correspondence in endless interval.