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Conditions of existence of extreme value laws and some formulae for the extremal index
The mere condition of ergodicity is not enough to have an extreme value law for a dynamical process. In fact, the conditions required must also be tied to the chosen observable as well. Indeed, even for systems with strong decay of correlations, if we chose an observable that is constant over the whole phase-space, one cannot expect to have a non-degenerate extreme value law associated to the process. Let us first refer to the classical (statistical) theory of EVT, where several results concerning the existence of an extreme value law under more or less strong mixing conditions have been found.
Classical conditions
First results concerning the existence of Extreme Value laws for processes with good dependence structure are due to Leadbetter [31]. Later, Chernick Hsing and McCormick [32] proved the existence of an extreme value law under some less strict hypotheses; the condition D(un) assuring asymptotic independence and the condition Dm(un), controlling the maximal order of clustering, that we are going to define now. Definition 3 We say that the condition D(un) is satisfied for the process X0, X1, … if for any integers i1 < … < ip and j1 < … < jk for which j1 − ip > t, and for all n > n0 ∈ N, |Fi1,…,ip,j1,…,jk (un) − Fi1…,ip (un)Fj1,…,jk (un)| ≤ α(n, t), (13) uniformly for every p, q ∈ N. Here, Fi1,…,ip is the joint distribution function of Xi1 , …, Xip , that is Fi1,…,ip (un) = µ(Xi1 < un, …, Xip < un).
EVT and the statistics of the number of visits
We mentioned in section (3.1) that θ quantifies the tendency of the process to form clusters of high values. We give now a more precise meaning to this statement and see that it appears in the asymptotic statistics of the number of visits of the system to the set Ωn. It is known from the classical statistical theory of extreme values that the number of exceedances of a process having an extreme value law, up to a re-scaled time, over a large threshold is given by a compound Poisson distribution [85]. Thanks to relation (15), exceedances over the high threshold un correspond in this context to visits of the system in the set Ωn. Let us describe more in details the objects and the results. Definition 7 We say that the integer valued random variable P follows a Poisso-nian distribution P if λke−λ P (P = k) = , (26) k! for some parameter λ.
Definition 8 We say that the integer valued random variable W is compound Poisson distributed if there exists
• Some integer valued i.i.d. random variables (Xj )j=1,2,… with the same prob-ability distribution given by πl = P(Xj = l), for l ∈ N
• A Poisson distributed random variable P such that P W =Xj i=1.
Numerical estimates of the extremal index
Let us consider a finite observational time series X0, …, Xn−1 which is obtained for example as in section (3.1), by computing the value taken by an observable along a trajectory. Although the extremal index is determined in terms of exceedances over some sequence of high thresholds, in practice we need to fix a single high threshold to define exceedances. Usually, a high quantile of the distribution 0 < ρ < 1 is taken, assuring at the same time that enough data are at disposal for the statistics. In some situations, estimations of the extremal index at the order 0 (involving only an estimate of 1 − p0) suffice. This is the case of systems satisfying the property D2(un). Under this assumption, the maximum likelihood estimate of S¨uveges [100] can be used. It is given by:
Extreme Value Theory and local dynamical in-dicators
The methods that we will present give access the fine geometrical structure of the attractor by computing its local dimension. We now present this quantity along with other dimensional quantities associated with the invariant measure.
Fractal dimensions
Let (M, T, µ) be a dynamical system defined by the transformation T acting on the metric space M (usually of subset of Rn or of the torus Tn) that leaves the probability measure µ invariant. Diverse dimensional quantities have been intro-duced to describe the geometrical structure of the measure, which can be extremely complex.
Local dimensions
To characterize the local scaling behavior of µ, Young [76] introduced the concept of local dimension. Definition 9 We call the local dimension d(z) of µ at the point z ∈ M the quantity defined by the following limit (if it exists): d(z) := lim log µ(B(z, r)) , (41) log r r→0 where B(z, r) denotes a ball centered in z of radius r. d(z) gives the scaling of the measure of a ball centered in z when its radius r goes to 0. For example, if we consider a transformation preserving the Lebesgue measure defined on the unit square, a point z in the interior of the square and denote dist(., .) the Euclidean distance, we have that d(z) = limr→0 log πr2 = 2, log r which matches our usual concept of dimensions. For measures living on more complex fractal supports or with non uniform densities, d(.) can also take non-integer values. For reasons that will be developed throughout the thesis, we also introduce the following quantity when the limit in equation (41) is not taken:
Table of contents :
1 Notations and basic definitions
2 Introduction
3 Extreme value theory for dynamical systems
3.1 General framework
3.2 Conditions of existence of extreme value laws and some formulae for the extremal index
3.2.1 Classical conditions
3.2.2 The spectral approach
3.3 EVT and the statistics of the number of visits
3.4 Numerical estimates of the extremal index
4 Extreme Value Theory and local dynamical indicators
4.1 Fractal dimensions
4.1.1 Local dimensions
4.1.2 Generalized dimensions
4.2 Local dimensions and inverse persistence
4.3 Local dimensions and generalized dimensions
4.4 Influence of noise on θz and on the local dimensions
4.4.1 I.i.d. random tranformations
4.4.2 Moving target
5 Extreme value theory and global dynamical indicators
5.1 A new observable
5.2 Generalized dimensions and the Block Maxima approach
5.3 The dynamical extremal index and the hyperbolicity index of order q
5.4 Influence of noise on the global dynamical indicators
5.4.1 Influence of uniform noise on the Dq spectrum
5.4.2 Influence of uniform noise on the DEI spectrum
5.4.3 Influence of discrete noise for the DEI spectrum
6 Extreme value theory and the statistics of visits in small target sets
6.1 Statistics of returns for some non-deterministic stationary systems .
6.2 Statistics of returns for climate data
6.3 Statistics of returns for sequential systems
6.4 Statistics of returns in the neighborhood of the diagonal
7 Generalized dimensions and large deviations of local dynamical quantities
7.1 Elements of large deviations theory
7.2 Generalized dimensions and large deviations of the local dimension
7.3 Generalized dimensions and large deviations of first return/hitting times
8 Further methods to compute the generalized dimensions using scaling relations
8.1 Using the Correlation Integral
8.2 Using return times
8.3 Using hitting times
8.4 Numerical extraction of the generalized dimensions
9 An application of EVT to the study of a real world multi-dimensional dynamical system; collective events in neural networks
9.1 Motivations
9.2 Neuronal avalanches
9.3 The dynamical model: a sparse network of Rulkov maps
9.4 Unperturbed dynamics of the single neuron
9.5 Perturbed dynamics of the single neuron: basin of attraction
9.6 Network topology and synaptic coupling
9.7 Network dynamics
9.7.1 Invariant measure
9.7.2 Firing cascades as trees of causal excitations
9.7.3 Statistical analysis of firing cascades
9.8 Statistics of extreme events in a finite time–interval
10 Conclusions
