带限制赌注的有效鞅
Effective Martingales with Restricted Wagers
摘要：
可计算随机性的经典模型考虑取实数或有理值的鞅。Bienvenu et al.（2012）和Teutsch（2014）最近的工作表明，当鞅取整数值时，经典模型的基本特征会发生变化。我们比较的预测能力的鞅的赌注属于三个不同的子集的有理数：（a）所有的有理数，（b）有理数不包括一个穿孔的邻域为0，（c）整数。我们还考虑了三个不同的成功标准：（i）积累无限数量的货币，（ii）消费无限数量的货币，（iii）使积累的资本振荡。（a）-（c）和（i）-（iii）的九个组合定义了九个可计算随机性的概念。我们提供了一个完整的表征这些概念之间的关系，并表明它们形成五个线性有序类。我们的结果解决了Bienvenu et al.（2012），Teutsch（2014）和Chalcraft et al.（2012）提出的突出问题，并加强了现有的结果。
英文摘要：
The classic model of computable randomness considers martingales that take real or rational values. Recent work by Bienvenu et al. (2012) and Teutsch (2014) shows that fundamental features of the classic model change when the martingales take integer values. We compare the prediction power of martingales whose wagers belong to three different subsets of rational numbers: (a) all rational numbers, (b) rational numbers excluding a punctured neighbourhood of 0, and (c) integers. We also consider three different success criteria: (i) accumulating an infinite amount of money, (ii) consuming an infinite amount of money, and (iii) making the accumulated capital oscillate. The nine combinations of (a)--(c) and (i)--(iii) define nine notions of computable randomness. We provide a complete characterization of the relations between these notions, and show that they form five linearly ordered classes. Our results solve outstanding questions raised in Bienvenu et al. (2012), Teutsch (2014), and Chalcraft et al. (2012), and strengthen existing results.
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。
