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Register a new userExtreme Value Laws for sequences of intermittent maps
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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}.
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In this paper we prove the existence of Extreme Value Laws for dynamical systems perturbed by instrument-like-error, also called observational noise. An orbit perturbed with observational noise mimics the behavior of an instrumentally recorded time series. Instrument characteristics - defined as precision and accuracy - act both by truncating and randomly displacing the real value of a measured observable. Here we analyze both these effects from a theoretical and numerical point of view. First we show that classical extreme value laws can be found for orbits of dynamical systems perturbed with observational noise. Then we present numerical experiments to support the theoretical findings and give an indication of the order of magnitude of the instrumental perturbations which cause relevant deviations from the extreme value laws observed in deterministic dynamical systems. Finally, we show that the observational noise preserves the structure of the deterministic attractor. This goes against the common assumption that random transformations cause the orbits asymptotically fill the ambient space with a loss of information about any fractal structures present on the attractor.
Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase--space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical r^ole of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non--standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.
Using quantitative perturbation theory for linear operators, we prove spectral gap for transfer operators of various families of intermittent maps with almost constant potentials (high-temperature regime). Holder and bounded p-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau-Manneville map, any potential with Lispchitz constant less than 0.0014 has a transfer operator acting on Lip([0, 1]) with a spectral gap; and that for any 2-to-1 unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on BV([0, 1]) with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in (Giulietti et al. 2015), allowing all results there to be applied under the high temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.
For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^ntoR$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation properties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.
We derive bounds on the distribution function, therefore also on the Value-at-Risk, of $varphi(mathbf X)$ where $varphi$ is an aggregation function and $mathbf X = (X_1,dots,X_d)$ is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of $mathbf X$, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Frechet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of $mathbf X$ is known on a subset of its domain, and finally we consider the case where the copula of $mathbf X$ lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Frechet--Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Frechet--Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.
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