Long range dependence: The effect of infinite ergodic theoretical structures on limit theorems in probability

长程依赖性：无限遍历理论结构对概率极限定理的影响

• 批准号: 1506783
• 负责人: Gennady Samorodnitsky
• 金额: $ 30万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1506783&HistoricalAwards=false
• 关键词: Long; range; dependence; effect; infinite

This is a project on stochastic models with heavy tails and long range dependence. Heavy tails appear when researchers need to model phenomena where extreme values appear relatively frequently, such as weather, seismology, finance, social networks, and others. Long range dependence appears when the memory persists in the system for a long time. This happens in economics, hydrology, supply chain management, and other areas. Both heavy tails and long range dependence affect in a major way how the forces in a system interact and combine. Failure to account appropriately for the heavy tails and long range dependence can lead to very misleading conclusions about stability and reliability of the system - underestimating the risks in a financial network, or underestimating the bottlenecks in a communication network are only two examples. The interaction of forces in a system with heavy tails and long range dependence is the subject of the proposal. This project deals with several interlinked important parts of probability theory. The first one is that of long range dependence. This has been and remains an active area of research, into which this proposal introduces a number of new ideas. Further, limits theorems of the central limit type, as well as related types, are bread and butter of probability theory, and the present proposal is likely to introduce an entire new class of such theorems and relate them to the notion of long range dependence. Next, interplay between the probability theory and ergodic theory is also classic, but the present proposal introduces new directions of such interplay, particularly with infinite ergodic theory. The expected results are, of course, results in probability, but some of the results can be of interest in ergodic theory itself. Finally, understanding the structure of non-Gaussian self-similar processes, particularly those with stationary increments, remains an important and difficult question, and the present project is likely to introduce new and unexpected families of such processes and, hence, allow a new point of view on self-similarity.

这是一个关于具有重尾和长程依赖性的随机模型的项目。当研究人员需要对极端值相对频繁出现的现象（例如天气、地震学、金融、社交网络等）进行建模时，就会出现重尾现象。当内存在系统中持续存在很长时间时，就会出现长范围依赖。这种情况发生在经济学、水文学、供应链管理和其他领域。重尾和长距离依赖性都在很大程度上影响系统中的力量如何相互作用和结合。未能适当考虑重尾和长程依赖性可能会导致有关系统稳定性和可靠性的非常误导性的结论 - 低估金融网络中的风险，或低估通信网络中的瓶颈只是两个例子。该提案的主题是具有重尾和长程依赖性的系统中的力量相互作用。该项目涉及概率论的几个相互关联的重要部分。 第一个是长期依赖。这一直是并且仍然是一个活跃的研究领域，该提案引入了许多新想法。此外，中心极限类型以及相关类型的极限定理是概率论的基础，并且当前的提议可能会引入一类全新的此类定理，并将它们与长程依赖的概念联系起来。接下来，概率论和遍历理论之间的相互作用也是经典的，但目前的建议引入了这种相互作用的新方向，特别是无限遍历理论。当然，预期结果是概率结果，但某些结果可能与遍历理论本身有关。最后，理解非高斯自相似过程的结构，特别是那些具有平稳增量的过程，仍然是一个重要而困难的问题，并且当前的项目可能会引入此类过程的新的和意想不到的族，因此，允许一个新的点关于自相似性的看法。