Rates of convergence in limit theorems of probabilistic number theory

概率数论极限定理的收敛率

• 批准号: 5450490
• 负责人: Professor Dr. Josef G. Steinebach
• 金额: --
• 依托单位: Faculty of Physics and Mathematics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2005
• 资助国家: 德国
• 起止时间: 2004-12-31 至 2009-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5450490?language=en
• 关键词: Rates; convergence; limit; theorems; probabilistic

Classical problems of numBer theory, such as the mean-value theorem and the central limit theorem for additive and multiplicative functions, are well-studied. It is the aim of this project to provide rates of convergence results for these classical theorems. In doing so, probabilistic ideas of measuring rates of convergence will be essential tools for our investigations. In order to study a number theoretic problem via a probabilistic model, we use the idea of the Stone-Cech compactification of the set of natural numbers. Depending on an initial number theoretic problem, and a class of arithmetic functions involved, we arrive at a modell of either sums or products, respectively, of independent terms leading either to summation theory or to martingale theory in probability. Applying corresponding convergence rates estimates and other appropriate results form probability theory, we get an analogous rate of convergence result in number theory. The estimation of the accuracy of approximations will be a problem having its own interest and value. Among probabilistic measures for the rate of convergence are the Lévy distance, global versions of the central limit theorem, asymptotic expansions, the Ibragimov-Heyde method, the Marcinkiewicz-Zygmund law of large numbers, complete convergence, and others. Not only additive arithmetic functions are to be studied here, but also multiplicative and q-multiplicative ones.

数论中的经典问题，如加性和乘性函数的中值定理和中心极限定理，已得到很好的研究。这是这个项目的目的是提供这些经典定理的收敛速度的结果。在这样做时，测量收敛速度的概率思想将是我们研究的重要工具。为了通过概率模型研究数论问题，我们使用自然数集合的Stone-Cech紧化的思想。根据一个初始的数论问题，和一类算术函数，我们到达一个模型的总和或产品，分别导致求和理论或鞅理论的概率的独立条款。应用相应的收敛速度估计和概率论中的其他适当结果，我们得到了数论中类似的收敛速度结果。近似值精度的估计将是一个有其自身利益和价值的问题。收敛速度的概率度量包括Lévy距离、中心极限定理的全局形式、渐近展开、Marcinkiewicz-Zygmund大数定律、完全收敛等。这里不仅要研究加法算术函数，还要研究乘法算术函数和q-乘法算术函数。