Extremal Index, Hitting Time Statistics and periodicity

Abstract

We give conditions to prove the existence of an Extremal Index for general stationary stochastic processes by detecting the presence of one or more underlying periodic phenomena. This theory, besides giving general useful tools to identify the extremal index, is also tailored to dynamical systems. In fact, we apply this idea to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. As in the authors' previous works on this topic, the analogy of these laws in the context of hitting time statistics is explained and exploited extensively.

References in corpus (6)

• Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution
• Laws of rare events for deterministic and random dynamical systems
• Extreme value laws in dynamical systems under physical observables
• Sharp error terms for return time statistics under mixing conditions
• Generalized Extreme Value distribution parameters as dynamical indicators of Stability
• Poisson approximation for the number of visits to balls in nonuniformly hyperbolic dynamical systems

Cited by in corpus (31)

• Extremes and Recurrence in Dynamical Systems
• Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution
• Laws of rare events for deterministic and random dynamical systems
• The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics
• Towards a General Theory of Extremes for Observables of Chaotic Dynamical Systems
• Extreme value laws in dynamical systems under physical observables
• Speed of convergence for laws of rare events and escape rates
• Rare Events for the Manneville-Pomeau map
• Extreme value statistics for dynamical systems with noise
• Extremal dichotomy for uniformly hyperbolic systems
• Generalized Extreme Value distribution parameters as dynamical indicators of Stability
• Clustering of extreme events created by multiple correlated maxima
• Extreme Value Laws for dynamical systems with countable extremal sets
• Extreme Value laws for dynamical systems under observational noise
• On the computation of the extremal index for time series
• Shrinking targets and eventually always hitting points for interval maps
• Extreme value distributions of observation recurrences
• Hitting times and periodicity in random dynamics
• Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis
• Dynamical counterexamples regarding the Extremal Index and the mean of the limiting cluster size distribution
• Sampling local properties of attractors via Extreme Value Theory
• Extreme Value Laws for Superstatistics
• Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems
• Mixing properties in the advection of passive tracers via recurrences and extreme value theory
• Clustering indices and decay of correlations in non-Markovian models
• Extremes and extremal indices for level set observables on hyperbolic systems
• Extreme Value Theory for Piecewise Contracting Maps with Randomly Applied Stochastic Perturbations
• A trichotomy for hitting times and escape rates for a class of unimodal maps
• Almost sure convergence of maxima for chaotic dynamical systems
• Rare events, exponential hitting times and extremal indices via spectral perturbation
• A functional limit theorem for a dynamical system with an observable maximised on a Cantor set