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

9625552 NEY抽象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将次可加性论点的适用性扩展到马尔可夫可加过程和无限记忆链。这就产生了快速而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。用上述方法研究与无穷阶链有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动的特例来研究。我们将研究一个悬而未决的问题，即在大深度的吸收体中，高于固定能量的电子和光子之间的平衡。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、研究随机网络中的巨大超越、估计大系统(例如，保险公司)中的故障概率以及许多其他设置中。将开发和应用新的技术来研究这些概率随着观测次数的增加而减少的速度。第二个要研究的主题是电子-光子级联的模型，特别是超过给定能级的电子数与光子数的比率。

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.