Chung-Diaconis-Graham随机过程的乘法对称化
A multiplicatively symmetrized version of the Chung-Diaconis-Graham random process
摘要：
本文考虑$X_{n+1}=a_nX_n+b_n \pmod p$形式的随机过程，其中$p$是奇数，$X_0=0$，$(a_0,b_0), (a_1,b_1), (a_2,b_2),...$是独立同分布，$a_n$和$b_n$与$P(a_n=2)=P(a_n=(p+1)/2)=1/2$和$P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$无关。这可以被视为Chung、Diaconis和Graham的随机过程的乘法对称版本。本文表明，顺序$(\log p)^2$步骤足以使$X_n$接近均匀分布在整数mod $p$上的所有奇数$p$，而顺序$(\log p)^2$步骤是必要的$X_n$接近均匀分布在整数mod $p$上。
英文摘要：
This paper considers random processes of the form $X_{n+1}=a_nX_n+b_n \pmod p$ where $p$ is odd, $X_0=0$, $(a_0,b_0), (a_1,b_1), (a_2,b_2),...$ 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 $p$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
