Complete convergence and records for dynamically generated stochastic processes

TL;DR

This paper studies the convergence behavior of multi-dimensional rare event point processes in dynamical systems, analyzing how clustering affects extremal process convergence and providing new formulas for point piling in bi-dimensional limits.

Contribution

It introduces a new formula for point piling in bi-dimensional processes with clustering and extends it to higher dimensions, advancing understanding of extremal processes in dynamical systems.

Findings

New formula for point piling in bi-dimensional processes with clustering

Complete convergence results for systems with fast decay of correlations

Clustering can prevent convergence of record times point processes

Abstract

We consider empirical multi-dimensional Rare Events Point Processes that keep track both of the time occurrence of extremal observations and of their severity, for stochastic processes arising from a dynamical system, by evaluating a given potential along its orbits. This is done both in the absence and presence of clustering. A new formula for the piling of points on the vertical direction of bi-dimensional limiting point processes, in the presence of clustering, is given, which is then generalised for higher dimensions. The limiting multi-dimensional processes are computed for systems with sufficiently fast decay of correlations. The complete convergence results are used to study the effect of clustering on the convergence of extremal processes, record time and record values point processes. An example where the clustering prevents the convergence of the record times point process is given.