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

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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.