Asymptotic theory for stochastic processes via martingale methods

建立相依序列极限定理的一个重要技巧是用熟知的结构（如鞅）来近似它们。平稳鞅逼近是随机过程的一个分支，在过去的十年里受到了广泛的关注。受非平稳过程和不能用平稳鞅逼近的平稳过程的渐近性质的研究的启发，我们将发展非平稳鞅逼近理论，它将项目的所有部分结合起来。新方法将利用分块技术来打破依赖性和一种新型的鞅结构的基础上块的变量。我们还将提供最大不等式，包括罗森塔尔型不等式，这是重要的渐近结果的收敛速度，并通过近似随机过程的研究，在几乎肯定的意义上，与独立的正常随机变量的总和。这些工具是基本的随机过程，确保渐近结果，包括条件功能的中心极限定理，极限定理开始于一个点，中等和大偏差的结果，以及精确表示的尾概率的总和随机变量获得新的投影准则。这些类型的渐近行为是概率论的核心，在统计学和其他应用领域有重要的应用。 该项目有望提供新的数学思想和技术，为随机过程的几个困难的开放问题提供新的思路，并将影响其他研究领域，如下：在一个点开始的极限定理对分析随机环境中的随机游动是有用的。他们也感兴趣的研究人员在统计力学，物理学，并将导致新的发现，一些有趣的间歇性地图，最近来到专家的注意力在动力系统。这些结果将适用于Metropolis-Hastings算法的家族，这些算法对于贝叶斯统计是必不可少的。尾概率的精确渐近表示将有助于在许多应用领域中以自然方式发生的偏差概率的估计，例如，在风险理论和金融中，在大额索赔保险的背景下的保险问题。研究结果将通过在顶级期刊上发表文章以及在国家和国际会议上发表演讲的方式广泛传播。他们还将与研究生的培训相结合，在每周的研讨会上提出，并将进入随机过程极限理论课程的课程。美国和欧洲的几个研究小组正在积极研究相关问题，拟议的项目将有助于加强国际科学交流和合作。

An important technique for establishing limit theorems for dependent sequences is to approximate them with well understood structures, such as martingales. Stationary martingale approximation is a subfield of stochastic processes that received a lot of attention in the last decade. Motivated by the study of asymptotic properties of nonstationary processes and also of stationary processes that cannot be approximated by stationary martingales, we shall develop the theory of nonstationary martingale approximations, which unites all the parts of the project. The new method will exploit blocking techniques to break the dependence and a new type of martingale construction based on blocks of variables. We shall also provide maximal inequalities, including Rosenthal-type inequalities, which are important for obtaining rates of convergence in the asymptotic results and facilitate the study of a stochastic process by approximating it, in the almost sure sense, with sums of independent normal random variables. These tools are fundamental for obtaining new projective criteria for stochastic processes that insure asymptotic results, including the conditional functional central limit theorem, limit theorems started at a point, moderate and large deviation results as well as exact representations for the tail probabilities of sums of random variables. These types of asymptotic behaviors are at the heart of probability theory with important applications to statistics and other applied fields. The proposed project is expected to provide new mathematical ideas and techniques that will shed new light on several difficult open problems for stochastic processes and will impact other fields of research as follows: The limit theorems started at a point are useful to analyze random walks in a random environment. They are also of interest to researchers working in statistical mechanics, physics, and will lead to new discoveries for some interesting intermittent maps that recently came to the attention of specialists in dynamical systems. The results will be applicable to families of Metropolis-Hastings algorithms that are essential, for instance, for Bayesian statistics. The exact asymptotic representation for the tail probabilities will facilitate the estimations of deviation probabilities that occur in a natural way in many applied areas, so for instance, in problems of insurance in the context of large claim insurance, in risk theory and finance. The results will be disseminated broadly through publications in top rated journals and presentations in national and international conferences. They will also be integrated with training of graduate students, presented in weekly seminars, and will enter the curriculum of a course on limit theory for stochastic processes. Related questions are actively being studied by several groups of researchers, in the US and in Europe, and the proposed project will contribute to strengthen international scientific exchange and collaborations.