Random Process Under The View Of Continued Fraction

In this paper, some original work has been made to continued fractions un-der the standpoint of stochastic process. Based on the well-known results from the ergodic theory of continued fractions,we point out that the continued fractions could be viewed as a stable、inversible stochastic process with each random vari-able identically distributed but not independent and therefore a new method for generating continued fractions has been created. Our method has its advantages over the original theoretical Gauss Map method in the real world due to the lim-itation of computer precision.The invention of the new method lays a foundation for stochastic stimulation on continued fractions and as an application, we study the renewal-process of continued fractions. It turns out that the beautiful results on i.i.dmodel which has recently been proved by Jian-Sheng Xie and Jiangang Y-ing could be applied into continued fractions as expected,and by doing this it also shows our method is efficient.The main result is following as expected:Rn=O(n1/2), limn�'∞(Rn,k)/(Rn)=(γΓ(k-γ))/(k!Γ(1-γ)),whereγ=1/2.