Extremes and Recurrence in Dynamical Systems

Abstract

ii 4.1.2 The New Conditions 44 4.1.3 The Existence of EVL for General Stationary Stochastic Processes under Weaker Hypotheses 46 4.1.4 Proofs of Theorem 4.1.4 and Corollary 4.1.5 48 4.2 Extreme Values for Dynamically Defined Stochastic Processes 53 4.2.1 Observables and Corresponding Extreme Value Laws 55 4.2.2 Extreme Value Laws for Uniformly Expanding Systems 59 4.2.3 Example 4.2.1 revisited 61 4.2.4 Proof of the Dichotomy for Uniformly Expanding Maps 63 4.3 Point Processes of Rare Events 64 4.3.1 Absence of Clustering 64 4.3.2 Presence of Clustering 65 4.3.3 Dichotomy for Uniformly Expanding Systems for Point Processes 67 4.4 Conditions Д q (u n ), D 3 (u n ), D p (u n ) * and Decay of Correlations 68 4.5 Specific Dynamical Systems where the Dichotomy Applies 71 4.5.1 Rychlik Systems 72 4.5.2 Piecewise Expanding Maps in Higher Dimensions 73 4.6 Extreme Value Laws for Physical Observables 74

Return levels of temperature extremes in southern Pakistan

“…Under very general conditions, the exceedances are asymptotically distributed according to the generalized Pareto distribution (GPD). The GPD has remarkable properties of universality when the asymptotic behavior is considered (Lucarini et al, 2016), while one can expect that the threshold level above which the asymptotic behavior is achieved depends on the characteristics of the analyzed time series. In particular, when looking at spatial fields, the threshold level depends on the geographical location.…”

“…While the POT method has been applied for studying temperature extremes in different regions of the world (Burgueño et al, 2002;Nogaj et al, 2006;Coelho et al, 2007;Ghil et al, 2011), to our knowledge, it has never been used to analyze the statistics of temperature extremes in Sindh. Thanks to the properties of universality of the GPD distribution (Lucarini et al, 2016), the POT approach can in principle provide reliable estimates of return periods and also the return levels (RLs) for time ranges longer than what is actually observed. This information and this predictive power can be beneficial for policy-makers and other stakeholders since it is exactly the kind of information planners need when designing infrastructures that must last a very long time.…”

“…We have done this because the statistical inference provided by the POT method provides a more efficient use of data and has better properties of convergence when finite datasets are considered with respect to alternative methods for the analysis of extremes, such as the block maxima method, which is used to fit the observed data to the generalized extreme value (GEV) distribution (Lucarini et al, 2016). Additionally, here we are interested in investigating the actual tails of the distributions and not the statistics of yearly maxima, for example.…”

An Interdisciplinary Approach to the Study of Extreme Weather Events: Large-Scale Atmospheric Controls and Insights from Dynamical Systems Theory and Statistical Mechanics

“…In meteorology, extreme events are often defined as very large/small values of an observable relative to a threshold, following the approach of Pickands (1975). In dynamical systems theory, this definition is extended to all rare states of the system (Lucarini et al 2016), which may not always be associated with very large/small values of a given variable. From a societal viewpoint, the extreme nature of an event is primarily linked to its impact on human lives, property, and infrastructure.…”

Predictability of Extreme Waves in the Lorenz-96 Model Near Intermittency and Quasi-Periodicity

“…Geophysical applications, in which dynamical systems arise as models and observables are physical quantities like wind speed or temperature, form an important motivation for the development of the theory. Very recently, Lucarini et al [8] published the first text book on extremes in dynamical systems which gives an excellent overview of the latest developments and also provides an extensive source of references.…”