Related papers: Rare events and Poisson point processes

Compound Poisson distributions and signed compound Poisson measures are used for approximation of the Markov binomial distribution. The upper and lower bound estimates are obtained for the total variation, local and Wasserstein norms. In a…

The probability of rare and extreme events is an important quantity for design purposes. However, computing the probability of rare events can be expensive because only a few events, if any, can be observed. To this end, it is necessary to…

Given an arbitrary long but finite sequence of observations from a finite set, we construct a simple process that approximates the sequence, in the sense that with high probability the empirical frequency, as well as the empirical one-step…

We present sufficient conditions for sums of dependent point processes to converge in distribution to a Poisson process. This extends the classical result of Grigelionis [Theory Probab. Appl. 8 (1963) 172--182] for sums of uniformly null…

We consider $N$ events that are defined on a common probability space. Those events shell have a common probability function that is symmetric with respect to interchanging the events. We ask for the probability distribution of the number…

In this article, we consider Poisson and Poisson convoluted geometric approximation to the sums of $n$ independent random variables under moment conditions. We use Stein's method to derive the approximation results in total variation…

It is well known that a binomial $(n,p)$ can be approximated by a Poisson distribution with parameter $np$. The typical approach in undergraduate probability texts is to show a convergence result for the distribution of the binomial as $n$…

If you take a superposition of n IID copies of a point process and thin that by a factor of 1/n, then the resulting process tends to a Poisson point process as n tends to infinity. We give a simple proof of this result that highlights its…

In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than…

Non-uniform estimates are obtained for Poisson, compound Poisson, translated Poisson, negative binomial and binomial approximations to sums of of m-dependent integer-valued random variables. Estimates for Wasserstein metric also follow…

Rare events refer to qualitatively unlikely events whose realization can nevertheless have important consequences. Typically, the prediction of the kinetics of these events relies on Arrhenius laws, with exponentially distributed waiting…

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…

The Poisson process is the most elementary continuous-time stochastic process that models a stream of repeating events. It is uniquely characterised by a single parameter called the rate. Instead of a single value for this rate, we here…

As an application of Stein's method for Poisson approximation, we prove rates of convergence for the tail probabilities of two scan statistics that have been suggested for detecting local signals in sequences of independent random variables…

Project a collection of points on the high-dimensional sphere onto a random direction. If most of the points are sufficiently far from one another in an appropriate sense, the projection is locally close in distribution to the Poisson point…

This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of Stein's equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson…

The evaluation of the probability of union of a large number of independent events requires several combinations involving the factorial and the use of high performance computers with several hours of processing. Bounds and simplifications…

We present a process-level Poisson-approximation result for the degree-k vertices in a high-density weighted random connection model with preferential-attachment kernel in the unit volume. Our main focus lies on the impact of the left tails…

This paper deals with Poisson approximation to weighted sums of locally dependent random variables using Stein's method. The derived result represents a significant improvement of existing results. To illustrate the effectiveness of our…

In this paper we use a Malliavin-Stein type method to investigate Poisson and normal approximations for the measurable functions of infinitely many independent random variables. We combine Stein's method with the difference operators in…