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程相关的随机模型的项目。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、金融、社交网络等，就会出现重尾。当记忆在系统中持续很长时间时，就会出现长程相关。这种情况发生在经济学、水文学、供应链管理和其他领域。重尾效应和长程相关性都对系统中各种力如何相互作用和联合收割机产生重大影响。如果不适当地考虑重尾和长程相关性，可能会导致对系统稳定性和可靠性的非常误导的结论-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。该提案的主题是在具有重尾和长程相关性的系统中力的相互作用。这个项目涉及概率论的几个相互关联的重要部分。 第一个是长期依赖性。这一直是而且仍然是一个活跃的研究领域，本提案在其中提出了一些新的想法。此外，中心极限类型的极限定理，以及相关类型，是概率论的面包和黄油，本提案可能会引入一个全新的一类这样的定理，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，但本建议引入了这种相互作用的新方向，特别是与无限遍历理论。当然，预期的结果是概率的结果，但有些结果可能是遍历理论本身的兴趣。最后，理解非高斯自相似过程的结构，特别是那些有固定增量的过程，仍然是一个重要而困难的问题，本项目可能会引入新的和意想不到的家庭，这样的过程，因此，允许一个新的观点自相似。

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.