Several Questions in Probability Theory

9503375 T. P. Hill摘要 本研究计画涉及数个与机率数学理论相关的主题，包括：最佳停止理论-- 停止随机过程以实现诸如最大化停止时过程的期望值的某个目标，重点在于基于不完全信息找到好的停止规则;划分问题--对二分的研究（例如，火腿三明治定理），公平划分（切蛋糕不等式）或一般划分（Lyapounov 凸性）的一个对象，以及应用到假设检验和博弈论;经验分布理论-估计一个未知的分布的基础上，从该分布的随机样本的研究，重点是新的替代权重函数;收敛和不变性原则-的措施和他们的关系，以限制法律的代表性的研究;有效位问题--研究经验数据有效位的非均匀性，及其在数学建模和计算机设计中的应用。 该项目主要是理论性的，但在统计学，经济学和计算机科学以及纯数学的其他领域（如测度，积分和函数论）中具有潜在的应用。 这个研究项目涉及与概率论有关的数学中的几个主题。 这些主题包括：控制或停止随机过程的理论方面，如污染水平 波动或通货膨胀率;公平分配问题的理论和应用，其中一个对象，如一块土地或其他资产必须在几个当事人之间划分和分配;基于随机样本和观察估计未知人口分布的研究;以及关于随机过程长期行为的一般理论，如人口增长或全球温度波动。 该项目主要是理论性的，直接应用于统计和其他数学领域，并可能应用于其他科学领域，如经济学，运筹学和计算机技术。

9503375 T.P.Hill Abstract This research project involves several topics related to the mathematical theory of probability including: optimal stopping theory -- the study of stopping a stochastic process to achieve some goal such as maximizing the expected value of the process at the time of stopping, with emphasis on finding good stop rules based on incomplete information; partitioning problems -- the study of bisection (e.g., ham-sandwich theorems), fair-division (cake-cutting inequalities) or general partitioning (Lyapounov convexity) of an object as well as applications to hypothesis testing and game theory; empirical distribution theory -- the study of estimating an unknown distribution based on random samples from that distribution, with emphasis on new alternative weight functions; convergence and invariance principles -- the study of representations of measures and their relationships to limit laws; and the significant-digit problem -- the study of non-uniformity in significant digits of empirical data, and its applications to mathematical modelling and computer design. This project is mostly theoretical, but has potential applications in statistics, economics and computer science, as well as in other areas of pure mathematics such as measure, integration and function theory. This research project involves several topics in mathematics related to the theory of probability. These topics include: the theoretical aspects of controlling or stopping a random process such as pollution-level fluctuations or inflation rates; the theory and application of fair-division problems in which an object such as a parcel of land or other assets must be divided and distributed among several parties; the study of estimating the distribution of an unknown population based on random samples and observations; and general theories about long-range behavior of random processes such as population growth or global temperature fluctuations. This pro ject is primarily theoretical, with direct applications to statistics and other areas of mathematics, and potential applications to other areas of science such as economics, operations research, and computer technology.