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

这是一个关于具有重尾和长范围依赖的随机模型的项目。当研究人员需要对极端值出现相对频繁的现象进行建模时，例如天气、地震学、金融、社会网络等，就会出现重尾。当内存在系统中存在较长时间时，就会出现长期依赖。这种情况发生在经济学、水文学、供应链管理和其他领域。重尾和长距离依赖都在很大程度上影响着系统中的力如何相互作用和组合。如果不能适当地考虑重尾和长距离依赖，可能会导致关于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，这只是两个例子。在一个具有重尾和长距离依赖的系统中，力的相互作用是这个提议的主题。这个项目涉及概率论中几个相互关联的重要部分。第一个是长距离依赖。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，中心极限类型的极限定理以及相关类型的极限定理是概率论的主要内容，而目前的建议很可能会引入一类全新的这类定理，并将它们与长程相关的概念联系起来。其次，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。预期的结果，当然，是概率的结果，但一些结果可能对遍历理论本身感兴趣。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，目前的项目可能会引入这种过程的新的和意想不到的家族，因此，允许对自相似的新观点。

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.