Related papers: The compound Poisson distribution and return times…

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in 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…

In this note we discuss additional properties of mixed Poisson distributions. We discuss the convergence of mixed Poisson distributions to its mixing distribution for the scaling parameter tending to infinity. Moreover, we obtain a central…

In this paper we prove two results. First we show that dynamical systems with a $\phi$-mixing measure have in the limit Poisson distributed return times almost everywhere. We use the Chen-Stein method to also obtain rates of convergence.…

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…

We show that for all $\psi$-mixing shifts distributions of the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of all points are close either to Poisson or to compound Poisson distributions. We also describe…

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…

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 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…

The Poisson distribution is the probability distribution of the number of independent events in a given period of time. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as…

In this paper we study the distribution of hitting and return times for observations of dynamical systems. We apply this results to get an exponential law for the distribution of hitting and return times for rapidly mixing random dynamical…

In this article we survey properties of mixed Poisson distributions and probabilistic aspects of the Stirling transform: given a non-negative random variable $X$ with moment sequence $(\mu_s)_{s\in\mathbb{N}}$ we determine a discrete random…

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.…

It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that…

In this paper the generalization of the Poisson distribution is derived for the case when each consecutive event changes event rate. A simple formula for the probability of observing of a given number of events for the selected period of…

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…

We prove a quenched limiting law for random measures on subshifts at periodic points. We consider a family of measures $\{\mu_\omega\}_{\omega\in\Omega}$, where the `driving space' $\Omega$ is equipped with a probability measure which is…

Count data are omnipresent in many applied fields, often with overdispersion. With mixtures of Poisson distributions representing an elegant and appealing modelling strategy, we focus here on how the tail behaviour of the mixing…

A recurrence formula for absolute central moments of Poisson distribution is suggested.

We examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its Poisson-like limit. Such generalizations might be found in quantum optics with imperfect detection. We…