A novel method based on augmented Markov vector process for the time-variant extreme value distribution of stochastic dynamical systems enforced by Poisson white noise

Suppose (f, 𝒳, ν) is a dynamical system and ϕ : 𝒳 → ℝ is an observation with a unique maximum at a (generic) point in 𝒳. We consider the time series of successive maxima Mn(x) := max {ϕ(x),…,ϕ ◦ fn-1(x)}. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. In this paper, for certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d. random variables, the convergence rates depend on the rate of mixing and the recurrence time statistics. For a range of applications, including uniformly expanding maps, quadratic maps, and intermittent maps, we establish corresponding convergence rates. We also establish convergence rates for certain hyperbolic systems such as Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as Hénon maps.

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GENERALIZED EXTREME VALUE DISTRIBUTION PARAMETERS AS DYNAMICAL INDICATORS OF STABILITY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412502768 ◽

2012 ◽

Vol 22 (11) ◽

pp. 1250276 ◽

Cited By ~ 12

Author(s):

DAVIDE FARANDA ◽

VALERIO LUCARINI ◽

GIORGIO TURCHETTI ◽

SANDRO VAIENTI

Keyword(s):

Dynamical Systems ◽

Local Stability ◽

Extreme Value Distribution ◽

Extreme Value ◽

Local Information ◽

Value Distribution ◽

Generalized Extreme Value Distribution ◽

Extreme Value Statistics ◽

Distribution Parameters ◽

Insight Into

We introduce a new dynamical indicator of stability based on the Extreme Value statistics showing that it provides an insight into the local stability properties of dynamical systems. The indicator performs faster than others based on the iteration of the tangent map since it requires only the evolution of the original systems and, in the chaotic regions, gives further information about the local information dimension of the attractor. A numerical validation of the method is presented through the analysis of the motions in the Standard map.

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An alternative expression for stochastic dynamical systems with parametric Poisson white noise

Probabilistic Engineering Mechanics ◽

10.1016/j.probengmech.2012.12.006 ◽

2013 ◽

Vol 32 ◽

pp. 1-4 ◽

Cited By ~ 13

Author(s):

Xu Sun ◽

Jinqiao Duan ◽

Xiaofan Li

Keyword(s):

Dynamical Systems ◽

White Noise ◽

Stochastic Dynamical Systems ◽

Poisson White Noise ◽

Alternative Expression

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Extreme Value Statistics of Wind Speed Data by the POT and ACER Methods

Journal of Offshore Mechanics and Arctic Engineering ◽

10.1115/1.4001419 ◽

2010 ◽

Vol 132 (4) ◽

Cited By ~ 2

Author(s):

A. Naess ◽

E. Haug

Keyword(s):

Time Series ◽

Wind Speed ◽

Extreme Value Distribution ◽

Extreme Value ◽

Value Distribution ◽

Extreme Value Statistics ◽

Peaks Over Threshold ◽

Threshold Method ◽

Wind Speed Data ◽

Novel Method

The paper describes a novel method for predicting the appropriate extreme value distribution derived from an observed time series. The method is based on introducing a cascade of conditioning approximations to the exact extreme value distribution. This allows for a rational way of capturing dependence effects in the time series. The performance of the method is compared with that of the peaks-over-threshold method.

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