Extreme Value Laws for Dynamical Systems with Countable Extremal Sets

Abstract

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 M, i.e., the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set M 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 [FFT12, FFT15].

Extreme value distributions of observation recurrences

“…The presence of the observable imposes some natural conditions on the combined choice of f and T if we want to satisfy Eq. (5). For instance if f is locally constant in the neighborhood of the target point z and µ is not atomic in z, we see immediately that Eq.…”

“…Application to climate data shows a compound Poisson distribution, despite the relative modest length of the time series and the unavoidable approximations in their detection. (5) In Section 6 we consider what happens when the dynamical system and the observable are randomly perturbed. We show with analytical and numerical arguments, that if the perturbation of the map produces a smooth stationary measure or the observable changes randomly but staying prevalent, then the dimension of the image measure becomes integer.…”

Dynamical counterexamples regarding the extremal index and the mean of the limiting cluster size distribution

“…In [15], we introduced the EI in the dynamical setting and established a relation between the appearance of clustering and the existence of underlying periodic phenomena in the structure of the stochastic process. In fact, in the dynamical context, as observed, in [15] and in the subsequent papers [5, 6, 12], clustering is directly related with the periodicity of the maximal set , that is, the set of points where the observable achieves the global maximum. Hence, can be interpreted as the largest of the periods of the underlying periodic phenomena present in the stochastic process.…”

“…This formula for the finite time cluster size distribution was used first in [16] and explicitly written for the first time in [4]. It appeared subsequently in [5,6]. This formula was derived during the proof of the convergence of REPP, which was based on a blocking type of argument.…”

Clustering indices and decay of correlations in non-Markovian models

“…For a size q ∈ N and a level a > 0, let us define the sets U a = {X 0 > a} and A This is the probability of not observing another exceedance (of level a) up to time q given that we begin with the observation of an exceedance at time 0. This formula was used firstly by O'Brien and then by other authors for the extremal index (see for instance [10,13,12,16]). The value of q is determined by the observable U a and the decay of correlation properties of the process.…”