论文标题
类别中随机对象的时刻问题
The moment problem for random objects in a category
论文作者
论文摘要
概率理论中的时刻问题询问了何时存在具有特定时刻元组的独特度量的标准。我们研究了类别中随机对象的此问题的一种变体，其中矩具有的平均表达数量给固定对象。当时刻不会太快增长时，我们给出了与这些时刻分布存在的必要条件，表明存在这种唯一的措施，以时刻为量度给出了措施的公式，并证明了通过这些限制时刻接近该特定度量的措施。我们的结果适用于满足某些有限条件的类别，并提供了一个类似于第二个同构定理的条件，包括有限组，有限模块，有限环以及这些类别的许多变化的类别。这项工作是由数字理论的非亚伯cohen-lenstra-martinet程序激励的，该计划旨在计算作为随机数字最大的随机数字最大不受影响的扩展的GALOIS组的随机分布组的分布。
The moment problem in probability theory asks for criteria for when there exists a unique measure with a given tuple of moments. We study a variant of this problem for random objects in a category, where a moment is given by the average number of epimorphisms to a fixed object. When the moments do not grow too fast, we give a necessary and sufficient condition for existence of a distribution with those moments, show that a unique such measure exists, give formulas for the measure in terms of the moments, and prove that measures with those limiting moments approach that particular measure. Our result applies to categories satisfying some finiteness conditions and a condition that gives an analog of the second isomorphism theorem, including the categories of finite groups, finite modules, finite rings, as well as many variations of these categories. This work is motivated by the non-abelian Cohen-Lenstra-Martinet program in number theory, which aims to calculate the distribution of random profinite groups arising as Galois groups of maximal unramified extensions of random number fields.
