关于一类奇异单调函数

A kind of singular monotone function

奇异函数是一极为重要的奇异型随机变量的分布函数 ,被广泛研究。单调的奇异函数的实例和证明方法已得到许多有意义的结果。本文目的是通过马氏链在空间上的实现 ,利用非齐次马氏链强大数定律构造一类不减的奇异函数。作为推论得到的是一类严格递增的奇异函数。此构造方法依赖于马氏链的强律 ,而不是Cantor函数的变形。

Singular function is an important distribution function of the singular random variable. It is studied widely. There have been many meaningful results both in examples and in proof methods. In this paper, we will construct a sort of nondecreasing singular function by the realization of Markov Chain on the space and by using of strong law of large numbers for nonhomogencous Markov chain. As a corollary, a strictly increasing singular function is obtained. This method depends on the strong law of Markov chain, rather than the transform of Cantor function. Thus the type and proof method of monotone singular function is enriched greatly.