Mathematical Sciences: Limit Theory and Large Deviation Theory of Markov and Infinite Memory Chains with Applications to Physical Models

数学科学：马尔可夫和无限记忆链的极限理论和大偏差理论及其在物理模型中的应用

• 批准号: 9625552
• 负责人: Peter Ney
• 金额: $ 6万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625552&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Limit; Theory; Large

9625552 Ney ABSTRACT Research will be carried out in two general areas: large deviation theory, and cascade theory. In large deviation theory, regeneration and renewal methods will be used to extend the applicability of sub-additivity arguments to Markov additive processes and to infinite memory chains. This yields quick and direct methods to establish existence of a large deviation principle. New techniques for identifying the relevant rate functions will be investigated. Large deviation of the iterates of certain monotone maps related to the infinite order chains will be studies by the above methods. Electron-photon cascades will be studied as special cases of multi-type branching random walks. An open question on the balance between electrons and photons above a fixed energy at large depths of an absorber will be studied. Large deviation theory is the study of the estimation and approximation of very small probabilities. These arise in the study of transmission errors in information communication, in studying large exceedances in random networks, in estimating failure probabilities in large systems (e.g. insurance companies) and many other settings. New techniques will be developed and applied to study the rate at which these probabilities decrease as the number of observations increases. A second subject to be studied is a model for electron-photon cascades, in particular, the ratio of the number of electrons to photons above a given energy level.

9625552 Ney 摘要 研究将在两个一般领域进行：大偏差理论和级联理论。 在大偏差理论中，再生和更新方法将用于将子可加性论证的适用性扩展到马尔可夫加性过程和无限记忆链。 这产生了快速而直接的方法来确定大偏差原理的存在。 将研究识别相关速率函数的新技术。 通过上述方法将研究与无限阶链相关的某些单调映射的迭代的大偏差。 电子-光子级联将作为多类型分支随机游走的特殊情况进行研究。 将研究吸收体大深度处固定能量之上的电子和光子之间的平衡的悬而未决的问题。 大偏差理论是对非常小概率的估计和近似的研究。 这些问题出现在信息通信中的传输错误研究、研究随机网络中的大量超出、估计大型系统（例如保险公司）和许多其他环境中的故障​​概率中。 将开发并应用新技术来研究这些概率随着观测数量的增加而下降的速率。 要研究的第二个主题是电子-光子级联模型，特别是给定能级以上的电子与光子数量之比。