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

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，再生和更新方法将被用来扩展次可加性参数的适用性马尔可夫可加性过程和无限记忆链。 这产生了快速和直接的方法来建立大偏差原理的存在性。 将研究用于识别相关速率函数的新技术。 利用上述方法研究了与无限级链有关的某些单调映射迭代的大偏差。 电子-光子级联将作为多类型分支随机游动的特殊情况进行研究。 本文将研究在吸收体的大深度处，电子和光子在固定能量以上的平衡问题。 大偏差理论是对极小概率的估计和近似的研究。 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概率，估计大型系统（例如保险公司）和许多其他设置中的故障概率。 将开发和应用新技术来研究这些概率随观测次数增加而降低的速率。 第二个要研究的课题是电子-光子级联的模型，特别是在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.