63 research outputs found
Extreme values for Benedicks-Carleson quadratic maps
We consider the quadratic family of maps given by with
, where is a Benedicks-Carleson parameter. For each of these
chaotic dynamical systems we study the extreme value distribution of the
stationary stochastic processes , given by , for
every integer , where each random variable is distributed
according to the unique absolutely continuous, invariant probability of .
Using techniques developed by Benedicks and Carleson, we show that the limiting
distribution of is the same as that which would
apply if the sequence was independent and identically
distributed. This result allows us to conclude that the asymptotic distribution
of is of Type III (Weibull).Comment: 18 page
Extreme Value Laws in Dynamical Systems for Non-smooth Observations
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
Complete convergence and records for dynamically generated stochastic processes
We consider empirical multi-dimensional Rare Events Point Processes that keep
track both of the time occurrence of extremal observations and of their
severity, for stochastic processes arising from a dynamical system, by
evaluating a given potential along its orbits. This is done both in the absence
and presence of clustering. A new formula for the piling of points on the
vertical direction of bi-dimensional limiting point processes, in the presence
of clustering, is given, which is then generalised for higher dimensions. The
limiting multi-dimensional processes are computed for systems with sufficiently
fast decay of correlations. The complete convergence results are used to study
the effect of clustering on the convergence of extremal processes, record time
and record values point processes. An example where the clustering prevents the
convergence of the record times point process is given
Speed of convergence for laws of rare events and escape rates
We obtain error terms on the rate of convergence to Extreme Value Laws for a
general class of weakly dependent stochastic processes. The dependence of the
error terms on the `time' and `length' scales is very explicit. Specialising to
data derived from a class of dynamical systems we find even more detailed error
terms, one application of which is to consider escape rates through small holes
in these systems
Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems
We develop and generalize the theory of extreme value for non-stationary
stochastic processes, mostly by weakening the uniform mixing condition that was
previously used in this setting. We apply our results to non-autonomous
dynamical systems, in particular to {\em sequential dynamical systems}, given
by uniformly expanding maps, and to a few classes of random dynamical systems.
Some examples are presented and worked out in detail
Extreme Value Laws for sequences of intermittent maps
We study non-stationary stochastic processes arising from sequential
dynamical systems built on maps with a neutral fixed points and prove the
existence of Extreme Value Laws for such processes. We use an approach
developed in \cite{FFV16}, where we generalised the theory of extreme values
for non-stationary stochastic processes, mostly by weakening the uniform mixing
condition that was previously used in this setting. The present work is an
extension of our previous results for concatenations of uniformly expanding
maps obtained in \cite{FFV16}.Comment: To appear in Proceedings of the American Mathematical Society. arXiv
admin note: substantial text overlap with arXiv:1510.0435
Laws of rare events for deterministic and random dynamical systems
The object of this paper is twofold. From one side we study the dichotomy, in
terms of the Extremal Index of the possible Extreme Value Laws, when the rare
events are centred around periodic or non periodic points. Then we build a
general theory of Extreme Value Laws for randomly perturbed dynamical systems.
We also address, in both situations, the convergence of Rare Events Point
Processes. Decay of correlations against observables will play a central
role in our investigations
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