Related papers: The compound Poisson limit ruling periodic extreme…

Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the…

We show that for systems that allow a Young tower construction with polynomially decaying correlations the return times to metric balls are in the limit Poisson distributed. We also provide error terms which are powers of logarithm of the…

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the…

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…

We study quantitative recurrence to rare events in Countable Markov Shifts with recurrent potentials, focusing on return-time statistics to natural target sets for every point. In the positive recurrent case, return-time processes…

Any limiting point process for the time normalized exceedances of high levels by a stationary sequence is necessarily compound Poisson under appropriate long range dependence conditions. Typically exceedances appear in clusters. The…

We study the process of suitably normalized successive return times to rare events in the setting of infinite-measure preserving dynamical systems. Specifically, we consider small neighborhoods of points whose measure tends to zero. We…

We show that a compound Poisson distribution holds for scaled exceedances of observables $\phi$ uniquely maximized at a periodic point $\zeta$ in a variety of two-dimensional hyperbolic dynamical systems with singularities $(M,T,\mu)$,…

We consider the return times dynamics to Bowen balls for continuous maps on metric spaces which have invariant probability measures with certain mixing properties. These mixing properties are satisfied for instance by systems that allow…

It is well known that, under broad assumptions, the time-scaled point process of exceedances of a high level by a stationary sequence converges to a compound Poisson process as the level grows. The purpose of this note is to demonstrate…

We show that for planar dispersing billiards the return times distribution is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB measure. Since the Poincar\'e return map is piecewise smooth but becomes singular at the…

For non-uniformly hyperbolic dynamical systems we consider the time series of maxima along typical orbits. Using ideas based upon quantitative recurrence time statistics we prove convergence of the maxima (under suitable normalization) to…

The Extremal Index is a parameter that measures the intensity of clustering of rare events and is usually equal to the reciprocal of the mean of the limiting cluster size distribution. We show how to build dynamically generated stochastic…

We obtain quenched hitting distributions to be compound Poissonian for a certain class of random dynamical systems. The theory is general and designed to accommodate non-uniformly expanding behavior and targets that do not overlap much with…

We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the…

We consider expanding systems with invariant measures that are uniformly expanding everywhere except on a small measure set and show that the limiting statistics of hitting times for zero measure sets are compound Poisson provided the…

Given a periodic point $\omega$ in a $\psi$-mixing shift with countable alphabet, the sequence $\{S_{n}\}$ of random variables counting the number of multiple returns to shrinking cylindrical neighborhoods of $\omega$ is considered.…

Let $\{\xi(k), k \in \mathbb{Z} \}$ be a stationary sequence of random variables and let $\{S_n, n \in \mathbb{N}_+ \}$ be a transient random walk in the domain of attraction of a stable law. In the previous work \cite{Nicolas_Ahmad}, under…

Classical peaks over threshold analysis is widely used for statistical modeling of sample extremes, and can be supplemented by a model for the sizes of clusters of exceedances. Under mild conditions a compound Poisson process model allows…

We consider random dynamical systems on manifolds modeled by a skew product which have certain geometric properties and whose measures satisfy quenched decay of correlations at a sufficient rate. We prove that the limiting distribution for…