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Introduction to the Modern Theory of Dynamical Systems

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An Introduction To The Theory Of Point Processes

The Bootstrap and Edgeworth Expansion

An Introduction to Probability Theory and its Applications, Volume I

We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models The applications include climatic, seismic and socio-economic events, along with their prediction Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems Both these results have implications for the study and prediction of natural hazards and their human impacts

https://npg.copernicus.org/articles/18/295/2011/npg-18-295-2011.pdf

Extreme events: dynamics, statistics and prediction

We consider discrete time dynamical systems and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We also give applications of our theory to higher dimensional examples, for which we also obtain classical extreme value laws and exponential hitting time statistics (for balls). We extend these ideas to the subsequent returns to asymptotically small sets, linking the Poisson statistics of both processes.

/pdf/hitting-time-statistics-and-extreme-value-theory-3zubsxyonw.pdf

Hitting time statistics and extreme value theory

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area of extreme events, underlining its relevance for mathematics, natural sciences, engineering, and social sciences. After exploring the basics of the classical theory of extreme events, the book presents a careful examination of how a dynamical system can serve as a generator of stochastic processes, and explores in detail the relationship between the hitting and return time statistics of a dynamical system and the possibility of constructing extreme value laws for given observables. Explicit derivation of extreme value laws are then provided for selected dynamical systems. The book then discusses how extreme events can be used as probes for inferring fundamental dynamical and geometrical properties of a dynamical system and for providing a novel point of view in problems of physical and geophysical relevance. A final summary of the main results is then presented along with a discussion of open research questions. Finally, an appendix with software in Matlab programming language allows the readers to develop further understanding of the presented concepts.

/pdf/extremes-and-recurrence-in-dynamical-systems-27chviqgqs.pdf

Extremes and Recurrence in Dynamical Systems

Hitting Time Statistics and Extreme Value Theory

https://www.reading.ac.uk/web/FILES/maths/AnaFreitascom_Reading_2013_6.pdf

Jorge Milhazes Freitas

Papers

Hitting time statistics and extreme value theory

Extremes and Recurrence in Dynamical Systems

Hitting Time Statistics and Extreme Value Theory

The extremal index, hitting time statistics and periodicity

Extremes and Recurrence in Dynamical Systems: Lucarini/Extremes and Recurrence in Dynamical Systems