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

[摘要]本文将从大偏差理论和级联理论两大方面进行研究。在大偏差理论中，将使用再生和更新方法将子可加性论证扩展到马尔可夫可加过程和无限存储链。这提供了快速和直接的方法来确定大偏差原理的存在性。将研究确定有关速率函数的新技术。本文将用上述方法研究一类与无穷阶链相关的单调映射的迭代大偏差问题。电子-光子级联将作为多类型分支随机漫步的特例来研究。在吸收体的大深度处，在固定能量以上的电子和光子之间的平衡的一个悬而未决的问题将被研究。大偏差理论是研究极小概率的估计和近似。这些问题出现在研究信息通信中的传输错误、研究随机网络中的大型超出情况、估计大型系统（例如保险公司）的故障概率和许多其他环境中。将开发和应用新技术来研究这些概率随着观测次数的增加而降低的速率。要研究的第二个主题是电子-光子级联的模型，特别是给定能级以上的电子与光子的数量之比。

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.