Asymptotic theory for stochastic processes via martingale methods

通过鞅方法的随机过程渐近理论

• 批准号: 1208237
• 负责人: Magda Peligrad
• 金额: $ 15.03万
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208237&HistoricalAwards=false
• 关键词: Asymptotic; theory; stochastic; processes; via

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.

建立相依序列极限定理的一个重要技术是用人们熟知的结构来逼近它们，例如令。平稳鞅逼近是随机过程的一个子域，在过去的十年里受到了广泛的关注。出于对非平稳过程和平稳过程的渐近性质的研究，我们将发展非平稳鞅逼近理论，它统一了项目的所有部分。新方法将利用分块技术来打破依赖关系，并基于变量块构造一种新型的鞅结构。我们还将提供极大不等式，包括Rosenthal型不等式，它们对于获得渐近结果的收敛速度是重要的，并通过用独立正态随机变量和来逼近几乎必然意义上的随机过程来促进随机过程的研究。这些工具是获得确保渐近结果的随机过程的新射影准则的基础，包括条件泛函中心极限定理、在一点开始的极限定理、中等偏差和大偏差结果以及随机变量和的尾概率的精确表示。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着重要的应用。该项目有望提供新的数学思想和技术，为解决随机过程的几个难题提供新的思路和技术，并将对其他研究领域产生影响，如：从一点开始的极限定理对于分析随机环境中的随机游动是有用的。它们也引起了从事统计力学和物理学工作的研究人员的兴趣，并将导致一些有趣的间歇性地图的新发现，这些地图最近引起了动力系统专家的注意。这些结果将适用于Metropolis-Hastings算法家族，这些算法对于贝叶斯统计来说是必不可少的。尾概率的精确渐近表示将有助于估计在许多应用领域中以自然方式发生的偏差概率，例如，在大索赔保险的背景下的保险问题，在风险理论和金融中。将通过在排名靠前的期刊上发表出版物以及在国内和国际会议上发表演讲来广泛传播这一成果。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，拟议中的项目将有助于加强国际科学交流与合作。