Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

Abstract

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 sequential dynamical systems, both given by uniformly expanding maps and by maps with a neutral fixed point, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail. Contents 1. Introduction 1.1. The motivation and the dynamical setting 1.2. Extreme Value Laws for general non-stationary processes 2. A general result for extreme value laws for non-stationary processes 2.1. Preliminaries to the argument 2.2. The construction of the blocks 2.3. Final argument 3. Sequential Dynamical Systems 3.1. General presentation 3.2. Stochastic processes for sequential systems 3.3. Examples 4. EVT for the sequential systems: an example of uniformly expanding map 4.1. EVT for the β-transformation

Extreme Value Laws for sequences of intermittent maps

“…We use an approach developed in [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 [FFV16]. …”

“…In this section, we revise the general theory developed in [FFV16] in order to prove the existence of EVL for non-stationary processes, which is particularly suitable for application to processes arising from non-autonomous systems. However, since in our application there is no clustering of exceedances, we simplify the exposition by adapting the general conditions and setting to this particular case of absence of clustering.…”

“…This fact precluded its application in a dynamical setting. In [FFV16], the authors developed a more general theory, based on necessary adjustments to Leadbetter's conditions and a much more refined way of dealing with clustering, originally developed in [FFT12,FFT15], which, ultimately, allowed the application to non-autonomous dynamical systems.…”

“…We will use the theory established in [FFV16] to study sequential dynamical systems originated by the composition of intermittent maps. Sequential dynamical systems were introduced by Berend and Bergelson [BB84], as a non-stationary system in which a concatenation of maps is applied to a given point in the underlying space, and the probability is taken as a conformal measure, which allows the use of the transfer operator (Perron-Fröbenius) as a useful tool to quantify the loss of memory of any prescribed initial observable.…”

Extremes and Recurrence in Dynamical Systems

“…Notice that the process X k (·) = φ(f ω k • · · · • f ω 1 (·)), equipped with the probability P given by the Lebesgue measure Leb is not necessarily stationary nor independent. By using the theory of sequential β-transformations developed in [254], we can apply the generalisation of Extreme Value Theory to non-stationary sequences obtained in the first part of [253] and actually verify the adapted conditions Д q (u n ) and Д q (u n ). We can also obtain that, when ζ is periodic, explicit expressions for the EI.…”