von Neumann [(1951). Various techniques used in connection with random digits. National Bureau of Standards Applied Math Series 12: 36–38] introduced a simple algorithm for generating independent unbiased random bits by tossing a (possibly) biased coin with unknown bias. While his algorithm fails to attain the entropy bound, Peres [(1992). Iterating von Neumann's procedure for extracting random bits. The Annals of Statistics 20(1): 590–597] showed that the entropy bound can be attained asymptotically by iterating von Neumann's algorithm. Let $b(n,p)$ denote the expected number of unbiased bits generated when Peres’ algorithm is applied to an input sequence consisting of the outcomes of $n$ tosses of the coin with bias $p$ . With $p=1/2$ , the coin is unbiased and the input sequence consists of $n$ unbiased bits, so that $n-b(n,1/2)$ may be referred to as the cost incurred by Peres’ algorithm when not knowing $p=1/2$ . We show that $\lim _{n\to \infty }\log [n-b(n,1/2)]/\log n =\theta =\log [(1+\sqrt {5})/2]$ (where $\log$ is the logarithm to base $2$ ), which together with limited numerical results suggests that $n-b(n,1/2)$ may be a regularly varying sequence of index $\theta$ . (A positive sequence $\{L(n)\}$ is said to be regularly varying of index $\theta$ if $\lim _{n\to \infty }L(\lfloor \lambda n\rfloor )/L(n)=\lambda ^\theta$ for all $\lambda > 0$ , where $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$ .) Some open problems on the asymptotic behavior of $nh(p)-b(n,p)$ are briefly discussed where $h(p)=-p\log p- (1-p)\log (1-p)$ denotes the Shannon entropy of a random bit with bias $p$ .

中文翻译：

---

Peres随机数生成算法的渐近分析

冯·诺依曼 [(1951)。与随机数字相关的各种技术。国家标准局应用数学丛书12: 36–38] 介绍了一种简单的算法，通过抛掷（可能）具有未知偏差的有偏硬币来生成独立的无偏随机位。虽然他的算法未能达到熵界，但 Peres [(1992). 迭代冯诺依曼提取随机位的过程。统计年鉴20(1): 590-597] 表明熵界可以通过迭代冯诺依曼算法渐近地获得。让 $b(n,p)$ 表示当 Peres 算法应用于由以下结果组成的输入序列时生成的预期无偏位数 $n$ 有偏见地抛硬币 $p$ . 和 $p=1/2$ ，硬币是无偏的，输入序列由 $n$ 无偏位，因此 $nb(n,1/2)$ 可以称为佩雷斯算法在不知道时所产生的成本 $p=1/2$ . 我们表明 $\lim _{n\to \infty }\log [nb(n,1/2)]/\log n =\theta =\log [(1+\sqrt {5})/2]$ （在哪里 $\日志$ 是底的对数 $2$ )，再加上有限的数值结果表明 $nb(n,1/2)$ 可能是有规律变化的索引序列 $\θ$ . （正序 $\{L(n)\}$ 据说是有规律地变化的指数 $\θ$ 如果 $\lim _{n\to \infty }L(\lfloor \lambda n\rfloor )/L(n)=\lambda ^\theta$ 对所有人 $\lambda > 0$ ， 在哪里 $\l地板 x\r地板$ 表示不超过的最大整数 $x$ .) 关于渐近行为的一些开放问题 $nh(p)-b(n,p)$ 简要讨论在哪里 $h(p)=-p\log p- (1-p)\log (1-p)$ 表示带有偏差的随机位的香农熵 $p$ .