Speed of convergence to an extreme value distribution for non-uniformly hyperbolic dynamical systems

Mark Holland, Matthew Nicol
2015 Stochastics and Dynamics  
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 M n (x) := max{φ(x), . . ., φ • f n−1 (x)}. Recent works have focused on the distributional convergence of such maxima (under suitable normalization) to an extreme value distribution. In this article, for certain dynamical systems, we establish convergence rates to the limiting distribution. In contrast to the case of i.i.d
more » ... ndom 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 Anosov systems, and discuss convergence rates for non-uniformly hyperbolic systems, such as Hénon maps.
doi:10.1142/s0219493715500288 fatcat:xjsofog3ung25aza6qejd7z2vm