120 research outputs found
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
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
The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics
We prove that the distributional limit of the normalised number of returns to
small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical
systems is compound Poisson. The returns to small balls around a fixed point in
the phase space correspond to the occurrence of rare events, or exceedances of
high thresholds, so that there is a connection between the laws of Return Times
Statistics and Extreme Value Laws. The fact that the fixed point in the phase
space is a repelling periodic point implies that there is a tendency for the
exceedances to appear in clusters whose average sizes is given by the Extremal
Index, which depends on the expansion of the system at the periodic point.
We recall that for generic points, the exceedances, in the limit, are
singular and occur at Poisson times. However, around periodic points, the
picture is different: the respective point processes of exceedances converge to
a compound Poisson process, so instead of single exceedances, we have entire
clusters of exceedances occurring at Poisson times with a geometric
distribution ruling its multiplicity.
The systems to which our results apply include: general piecewise expanding
maps of the interval (Rychlik maps), maps with indifferent fixed points
(Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.Comment: To appear in Communications in Mathematical Physic
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
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