This paper considers random processes of the form \(X_{n+1}=a_nX_n+b_n\pmod p\) where p is odd, \(X_0=0\), \((a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots \) are i.i.d., and \(a_n\) and \(b_n\) are independent with \(P(a_n=2)=P(a_n=(p+1)/2)=1/2\) and \(P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3\). This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order \((\log p)^2\) steps suffice for \(X_n\) to be close to uniformly distributed on the integers mod p for all odd p while order \((\log p)^2\) steps are necessary for \(X_n\) to be close to uniformly distributed on the integers mod p.

本文考虑形式为\（X_ {n + 1} = a_nX_n + b_n \ pmod p \）的随机过程，其中p为奇数，\（X_0 = 0 \），\（（a_0，b_0），（a_1，b_1 ），（a_2，b_2），\ ldots \）是iid，并且\（a_n \）和\（b_n \）与\（P（a_n = 2）= P（a_n =（p + 1）/ 2）独立）= 1/2 \）和\（P（b_n = 1）= P（b_n = 0）= P（b_n = -1）= 1/3 \）。这可以看作是Chung，Diaconis和Graham的随机过程的乘对称形式。本文表明，为了\（（\的log P）^ 2 \）步骤足矣\（X_n \）为接近在整数模均匀分布p为所有奇数p为了使\（X_n \）接近于均匀分布在整数mod p上，必须使用阶数\（（（log p）^ 2 \）的步骤。