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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

https://www.fc.up.pt/pessoas/jmfreita/homeweb/publications/ExtDynSyst_FF-1-08.pdf

On the link between dependence and independence in extreme value theory for dynamical systems

We prove the equivalence between the existence of a non-trivial hitting time statistics law and Extreme Value Laws in the case of dynamical systems with measures which are not absolutely continuous with respect to Lebesgue. This is a counterpart to the result of the authors in the absolutely continuous case. Moreover, we prove an equivalent result for returns to dynamically defined cylinders. This allows us to show that we have Extreme Value Laws for various dynamical systems with equilibrium states with good mixing properties. In order to achieve these goals we tailor our observables to the form of the measure at hand.

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of certain non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

http://www.cpt.univ-mrs.fr/~vaienti/AFLV_final.pdf

Jorge Milhazes Freitas

Papers

On the link between dependence and independence in extreme value theory for dynamical systems

Extreme Value Laws in Dynamical Systems for Non-smooth Observations

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

From rates of mixing to recurrence times via large deviations

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