论文标题
Agafonov的有限和无限字母定理和概率分布不同于等分分布
Agafonov's Theorem for finite and infinite alphabets and probability distributions different from equidistribution
论文作者
论文摘要
字母$σ$上的无限序列$α$是$μ$分布的W.R.T.概率映射$μ$，如果对于每个有限字符串$ w $，则存在$ w $ $α$的限制频率，并且等于$μ（w）$。 ％我们提出了一个问题，即如何表征概率地图$μ$，该$ $ $ $ $ $分布在有限国家选择中保留，或同等地通过使用恒定空间进行程序选择。 We prove the following result for any finite or countably infinite alphabet $Σ$: every finite-state selector over $Σ$ selects a $μ$-distributed sequence from every $μ$-distributed sequence \emph{if and only if} $μ$ is induced by a Bernoulli distribution on $Σ$, that is a probability distribution on the alphabet extended to words by taking the product.我们主要结果的主要结果是对有限和无限字母的一组概率图的完整表征，其中有限状态选择保留了$μ$ $ $分布。主要的积极要点是，（适当的概括）Agafonov定理对有限和无数字母的Bernoulli分布（而不是等于等分分配）持有。进一步的结果，我们获得了符号动力学系统领域的结果：换档不变的测量$μ$上的$σ^ω$ $ $ $ $ $ $σ^ω$，以使任何有限态选择器都保留了$μ$的通用性属性，恰好是Bernoulli阳性的测量。
An infinite sequence $α$ over an alphabet $Σ$ is $μ$-distributed w.r.t. a probability map $μ$ if, for every finite string $w$, the limiting frequency of $w$ in $α$ exists and equals $μ(w)$. %We raise the question of how to characterize the probability maps $μ$ for which $μ$-distributedness is preserved across finite-state selection, or equivalently, by selection by programs using constant space. We prove the following result for any finite or countably infinite alphabet $Σ$: every finite-state selector over $Σ$ selects a $μ$-distributed sequence from every $μ$-distributed sequence \emph{if and only if} $μ$ is induced by a Bernoulli distribution on $Σ$, that is a probability distribution on the alphabet extended to words by taking the product. The primary -- and remarkable -- consequence of our main result is a complete characterization of the set of probability maps, on finite and infinite alphabets, for which finite-state selection preserves $μ$-distributedness. The main positive takeaway is that (the appropriate generalization of) Agafonov's Theorem holds for Bernoulli distributions (rather than just equidistributions) on both finite and countably infinite alphabets. As a further consequence, we obtain a result in the area of symbolic dynamical systems: the shift-invariant measures $μ$ on $Σ^ω$ such that any finite-state selector preserves the property of genericity for $μ$, are exactly the positive Bernoulli measures.
