Weak convergence to extremal processes and record events for non-uniformly hyperbolic dynamical systems

Abstract

For a measure-preserving dynamical system (X, ƒ, μ), we consider the time series of maxima Mn = max{X1,…,Xn} associated to the process Xn = φ (ƒn-1(x)) generated by the dynamical system for some observable φ : Χ → R . Using a point-process approach we establish weak convergence of the process Yn(t) = an(M[nt] - bn) to an extremal Y(t) process for suitable scaling constants an, bn ∈ R . Convergence here takes place in the Skorokhod space D(0, ∞) with the J1 topology. We also establish distributional results for the record times and record values of the corresponding maxima process.