Several Questions in Probability Theory

首席研究员正在开展与概率数学理论有关的课题研究。具体目标包括继续他对最优停止问题(特别是不完全信息下的停止)的研究，构造随机分布，本福德定律的分析和应用，一般的划分不等式，以及收敛和不变原理。在新的方向上，他正在研究期望极值和一般连续性表示定理的尖锐不等式。这项研究包括理论和应用两个部分，涉及与佐治亚理工学院和其他大学的其他研究人员以及几名研究生的合作。这项研究涉及对随机过程的研究，例如当存在不可预测的波动时，不断演变的金融市场或政府政策。如果对随机波动的性质有完全的了解，就能知道很多东西，但在实践中通常不是这样。这项研究的目的之一是试图发现在只有关于随机波动的部分信息可用时，决定何时停止或改变政策的好方法。这项研究的另一个主旨是开发新的统计工具来估计随机过程中的变化。这些想法的一个非常新的用途是应用于检测金融数据中的欺诈。这个想法是，捏造的数据通常与真实数据在性质上不同，最近才有基于这些波动理论的测试被证明在区分真实数据和捏造数据方面是有用的。随机测量研究的应用包括空间质量分布的新模型，包括物理和材料科学中的各种设置。

The principal investigator is carrying out research on topics related to the mathematical theory of probability. Specific objectives include continuation of his investigations of optimal-stopping problems (especially stopping under incomplete information), constructions of random distributions, analysis and applications of Benford's law, general partitioning inequalities, and convergence and invariance principles. In new directions he is studying sharp inequalities for expected extrema and general continuity-representation theorems. The research has both theoretical and applied components, and involves collaboration with other researchers both at Georgia Tech and other universities, as well as with several graduate students. This research involves the study of random processes, such as evolving financial markets or government policies, when unpredictable fluctuations are present. Much is known if there is complete knowledge of the nature of the random fluctuations, but this is usually not the case in practice. One of the thrusts of this research is to try to discover good methods for deciding when to stop or change a policy when only partial information is available about the random fluctuations. Another thrust of this research is to develop new statistical tools for estimating changes in random processes. One very new use of these ideas is application to detection of fraud in financial data. The idea is that fabricated data is usually qualitatively different from true data, and there are just recently tests based on these fluctuation theories which are proving useful in distinguishing true from fabricated data. Applications of the research on random measures includes new models for the distribution of mass in space, including various settings in physics and materials science.