Ana Cristina Moreira Freitas

The extremal index, hitting time statistics and periodicity

Abstract The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors’ previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.

Extreme values for Benedicks–Carleson quadratic maps

We consider the quadratic family of maps given by

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

f_{a}(x)=1-a x^2

Extremes and Recurrence in Dynamical Systems

with

Speed of convergence for laws of rare events and escape rates

x\in [-1,1]

Rare events for the Manneville–Pomeau map

, where

Extremal dichotomy for uniformly hyperbolic systems

a

Limiting behaviour of a geometric-type estimator for tail indices

is a Benedicks-Carleson parameter. For each of these chaotic dynamical systems we study the extreme value distribution of the stationary stochastic processes

New insights on the mitochondrial proteome plasticity in Parkinson's disease

X_0,X_1,...

Clustering of extreme events created by multiple correlated maxima

, given by