3,499 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
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 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
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
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
Rare Events for the Manneville-Pomeau map
We prove a dichotomy for Manneville-Pomeau maps : given
any point , either the Rare Events Point Processes (REPP),
counting the number of exceedances, which correspond to entrances in balls
around , converge in distribution to a Poisson process; or the point
is periodic and the REPP converge in distribution to a compound Poisson
process. Our method is to use inducing techniques for all points except 0 and
its preimages, extending a recent result by Haydn, Winterberg and Zweim\"uller,
and then to deal with the remaining points separately. The preimages of 0 are
dealt with applying recent results by Ayta\c{c}, Freitas and Vaienti. The point
is studied separately because the tangency with the identity map at
this point creates too much dependence, which causes severe clustering of
exceedances. The Extremal Index, which measures the intensity of clustering, is
equal to 0 at , which ultimately leads to a degenerate limit
distribution for the partial maxima of stochastic processes arising from the
dynamics and for the usual normalising sequences. We prove that using adapted
normalising sequences we can still obtain non-degenerate limit distributions at
Clustering of extreme events created by multiple correlated maxima
We consider stochastic processes arising from dynamical systems by evaluating
an observable function along the orbits of the system. The novelty is that we
will consider observables achieving a global maximum value (possible infinite)
at multiple points with special emphasis for the case where these maximal
points are correlated or bound by belonging to the same orbit of a certain
chosen point. These multiple correlated maxima can be seen as a new mechanism
creating clustering. We recall that clustering was intimately connected with
periodicity when the maximum was achieved at a single point. We will study this
mechanism for creating clustering and will address the existence of limiting
Extreme Value Laws, the repercussions on the value of the Extremal Index, the
impact on the limit of Rare Events Points Processes, the influence on
clustering patterns and the competition of domains of attraction. We also
consider briefly and for comparison purposes multiple uncorrelated maxima. The
systems considered include expanding maps of the interval such as Rychlik maps
but also maps with an indifferent fixed point such as Manneville-Pommeau maps
Extreme Value Laws for dynamical systems with countable extremal sets
We consider stationary stochastic processes arising from dynamical systems by
evaluating a given observable along the orbits of the system. We focus on the
extremal behaviour of the process, which is related to the entrance in certain
regions of the phase space, which correspond to neighbourhoods of the maximal
set , i.e. the set of points where the observable is maximised. The
main novelty here is the fact that we consider that the set may
have a countable number of points, which are associated by belonging to the
orbit of a certain point, and may have accumulation points. In order to prove
the existence of distributional limits and study the intensity of clustering,
given by the Extremal Index, we generalise the conditions previously introduced
in \cite{FFT12,FFT15}.Comment: arXiv admin note: text overlap with arXiv:1505.0155
Extremal dichotomy for uniformly hyperbolic systems
We consider the extreme value theory of a hyperbolic toral automorphism showing that if a H\"older observation
which is a function of a Euclidean-type distance to a non-periodic point
is strictly maximized at then the corresponding time series
exhibits extreme value statistics corresponding to an iid
sequence of random variables with the same distribution function as and
with extremal index one. If however is strictly maximized at a periodic
point then the corresponding time-series exhibits extreme value statistics
corresponding to an iid sequence of random variables with the same distribution
function as but with extremal index not equal to one. We give a formula
for the extremal index (which depends upon the metric used and the period of
). These results imply that return times are Poisson to small balls centered
at non-periodic points and compound Poisson for small balls centered at
periodic points.Comment: 21 pages, 4 figure
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