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

Abstract

For a measure-preserving dynamical system (X , f, µ), we consider the time series of maxima Mn = max{X1, . . . , Xn} associated to the process Xn = φ( f n−1 (x)) generated by the dynamical system for some observable φ : X → R. Using a pointprocess approach we establish weak convergence of the process Yn(t) = an(M[nt] − bn) to an extremal process Y (t) 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.