Convergence of Marked Point Processes of Excesses for Dynamical Systems

TL;DR

This paper studies the convergence of marked point processes derived from dynamical systems' extremal events, providing conditions for convergence to a compound Poisson process and explicit formulas for the multiplicity distribution.

Contribution

It establishes conditions for the convergence of marked point processes in dynamical systems and explicitly computes the limit distributions, including a generalized Pareto distribution.

Findings

Marked point processes converge to a compound Poisson process under decay of correlations.

Explicit formulas for the multiplicity distribution are derived.

Convergence results extend from induced maps to original systems.

Abstract

We consider stochastic processes arising from dynamical systems simply by evaluating an observable function along the orbits of the system and study marked point processes associated to extremal observations of such time series corresponding to exceedances of high thresholds. Each exceedance is marked by a quantity intended to measure the severity of the exceedance. In particular, we consider marked point processes measuring the aggregate damage by adding all the excesses over the threshold that mark each exceedance (AOT) or simply by adding the largest excesses in a cluster of exceedances (POT). We provide conditions to prove the convergence of such marked point processes to a compound Poisson process, for whose multiplicity distribution we give an explicit formula. These conditions are shown to follow from a strong form of decay of correlations of the system. Moreover, we prove that the convergence of the marked point processes for a `nice' first return induced map can be carried to the original system. The systems considered include non-uniformly expanding maps (in one or higher dimensions), maps with intermittent fixed points or non-recurrent critical points. For a general class of examples, the compound Poisson limit process is computed explicitly and, in particular, in the POT case we obtain a generalised Pareto multiplicity distribution.