Turning a Coin over Instead of Tossing It
Turning a Coin over Instead of Tossing It
复制标题
把硬币翻过来而不是扔掉
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
S. Volkov
中科院分区：
文献类型：
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作者：
J. Engländer;S. Volkov
给定 [0, 1] 中的数字序列 $$(p_n)_{nge 2}$$(pn)n≥2，请考虑以下实验。首先，我们翻转一枚公平的硬币，然后在步骤 n，我们以 $$p_n$$pn、$$nge 2$$n≥2 的概率将硬币翻转到另一面，与前面各项的顺序无关。关于正面朝上的经验频率的分布我们能说什么$$n
ightarrow infty $$n→∞?我们表明，随着转动变慢（即 $$p_n$$pn 变小），会发生许多相变，首先导致中心极限定理的崩溃，然后导致大数定律的崩溃。事实证明，临界状态是 $$p_n= ext {const}/n$$pn=const/n。在缩放限制中，我们获得均匀定律、高斯定律、半圆定律和反正弦定律。
Given a sequence of numbers $$(p_n)_{nge 2}$$(pn)n≥2 in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability $$p_n$$pn, $$nge 2$$n≥2, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as $$n
ightarrow infty $$n→∞? We show that a number of phase transitions take place as the turning gets slower (i. e., $$p_n$$pn is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is $$p_n= ext {const}/n$$pn=const/n. Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.