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

Abstract

For a measure-preserving dynamical system $({\mathcal{X}},f,\unicode[STIX]{x1D707})$, we consider the time series of maxima $M_{n}=\max \{X_{1},\ldots ,X_{n}\}$ associated to the process $X_{n}=\unicode[STIX]{x1D719}(f^{n-1}(x))$ generated by the dynamical system for some observable $\unicode[STIX]{x1D719}:{\mathcal{X}}\rightarrow \mathbb{R}$. Using a point-process approach we establish weak convergence of the process $Y_{n}(t)=a_{n}(M_{[nt]}-b_{n})$ to an extremal process $Y(t)$ for suitable scaling constants $a_{n},b_{n}\in \mathbb{R}$. Convergence here takes place in the Skorokhod space $\mathbb{D}(0,\infty )$ with the $J_{1}$ topology. We also establish distributional results for the record times and record values of the corresponding maxima process.