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

TL;DR

This paper proves the weak convergence of maxima processes derived from non-uniformly hyperbolic dynamical systems to extremal processes, and analyzes record times and values using a point process approach.

Contribution

It introduces a novel point process framework to establish weak convergence of maxima in non-uniformly hyperbolic systems, including record event analysis.

Findings

Weak convergence of maxima processes to extremal processes in Skorokhod space.

Distributional results for record times and record values.

Applicability to non-uniformly hyperbolic dynamical systems.

Abstract

For a measure preserving dynamical system $(\mathcal{X},f, \mu)$, we consider the time series of maxima $M_n=\max\{X_1,\ldots,X_n\}$ associated to the process $X_n=\phi(f^{n-1}(x))$ generated by the dynamical system for some observable $\phi:\mathcal{X}\to\mathbb{R}$. 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 taking 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.