Related papers: Estimating the Upcrossings Index
The upcrossings index $0\leq \eta\leq 1,$ a measure of the degree of local dependence in the upcrossings of a high level by a stationary process, plays, together with the extremal index $\theta,$ an important role in extreme events…
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…
The notion of multivariate upcrossings index of a stationary sequence ${\bf{X}}=\{(X_{n,1},\ldots,X_{n,d})\}_{n\geq 1}$ is introduced and its main properties are derived, namely the relations with the multivariate extremal index and the…
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…
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…
Given a simple transient random walk $(S_n)_{n\geq 0}$ in $\mathbf{Z}$ and a stationary sequence of real random variables $(\xi(s))_{s\in \mathbf{Z}}$, we investigate the extremes of the sequence $(\xi(S_n))_{n\geq 0}$. Under suitable…
We consider a random connection model (RCM) $\xi$ driven by a Poisson process $\eta$. We derive exponential moment bounds for an arbitrary cluster, provided that the intensity $t$ of $\eta$ is below a certain critical intensity $t_T$. The…
We establish Poisson and compound Poisson approximations for stabilizing statistics of $\beta$-mixing point processes and give explicit rates of convergence. Our findings are based on a general estimate of the total variation distance of a…
We consider stationary configurations of points in Euclidean space which are marked by positive random variables called scores. The scores are allowed to depend on the relative positions of other points and outside sources of randomness.…
We consider the clustering of extremes for stationary regularly varying random fields over arbitrary growing index sets. We study sufficient assumptions on the index set such that the limit of the point random fields of the exceedances…
The extremes of a stationary time series typically occur in clusters. A primary measure for this phenomenon is the extremal index, representing the reciprocal of the expected cluster size. Both a disjoint and a sliding blocks estimator for…
Asymptotic behavior of the point process of high and medium values of a Gaussian stationary process with discrete time is considered. An approximation by a Poisson cluster point process is given for the point process.
Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long- and short-range dependence of extremes may both appear. In…
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…
Stationary stochastic processes with independent increments, of which the Poisson process is a prominent example, are widely used to describe real world events. With the basic assumption that a counting process is stationary and has…
We consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$ obeying a very broad geometrical assumption on how the sequence expands. Under certain mixing and local conditions, we show how the tail…
A compound Poisson process whose randomized time is an independent Poisson process is called compound Poisson process with Poisson subordinator. We provide its probability distribution, which is expressed in terms of the Bell polynomials,…
The extremal index parameter theta characterizes the degree of local dependence in the extremes of a stationary time series and has important applications in a number of areas, such as hydrology, telecommunications, finance and…
Relying on the excursion set theory, we compute the number density of local extrema and crossing statistics versus the threshold for the stock market indices. Comparing the number density of excursion sets calculated numerically with the…
The extremal index is an important parameter in the characterization of extreme values of a stationary sequence. Our new estimation approach for this parameter is based on the extremal behavior under the local dependence condition…