Limiting Point Process Research Articles

Stable limits for sums of dependent infinite variance random variables

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parameters of the limit distribution are expressed in terms of some limiting point process. In this paper we will be able to determine the parameters of the limiting stable distribution in terms of some tail characteristics of the underlying stationary sequence. We will apply our results to some standard time series models, including the GARCH(1, 1) process and its squares, the stochastic volatility models and solutions to stochastic recurrence equations.

Asymptotic distributions for the intervals estimators of the extremal index and the cluster-size probabilities

Under appropriate long range dependence conditions, the point process of exceedances of a stationary sequence weakly converges to a homogeneous compound Poisson point process. This limiting point process can be characterized by the extremal index and the cluster-size probabilities. In this paper we address the problem of estimating these quantities and we consider the intervals estimators introduced in Ferro and Segers [2003. Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 545–556] and in Ferro [2004. Statistical methods for clusters of extreme values. Ph.D. Thesis, Lancaster University]. We establish asymptotic weak convergence to Gaussian random variables and we give their asymptotic variance.

Continuum limits of random matrices and the Brownian carousel

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general β siblings converge to Sine β , a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine β is continuous in the gap size and β, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at β=2.

Inference for the limiting cluster size distribution of extreme values

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 underlying Poisson points represent the cluster positions and the multiplicities correspond to the cluster sizes. In the present paper we introduce estimators of the limiting cluster size probabilities, which are constructed through a recursive algorithm. We derive estimators of the extremal index which plays a key role in determining the intensity of cluster positions. We study the asymptotic properties of the estimators and investigate their finite sample behavior on simulated data.

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X j,n }1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S n =∑ =1 X j,n to an infinitely divisible random variable whose Levy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.

DISTRIBUTION OF LOCALIZATION CENTERS IN SOME DISCRETE RANDOM SYSTEMS

As a supplement of our previous work [10], we consider the localized region of the random Schrödinger operators on l2(Zd) and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that limiting point processes are infinitely divisible.

Subsampling techniques and the Jackknife methodology in the estimation of the extremal index

For a sequence of independent, identically distributed random variables any limiting point process for the time normalized exceedances of high levels is a Poisson process. However, for stationary dependent sequences, under general local and asymptotic dependence restrictions, any limiting point process for the time normalized exceedances of high levels is a compound Poisson process, i.e., there is a clustering of high exceedances, where the underlying Poisson points represent cluster positions, and the multiplicities correspond to the cluster sizes. For such classes of stationary sequences there exists the extremal index θ , 0 ⩽ θ ⩽ 1 , directly related to the clustering of exceedances of high values. The extremal index θ is equal to one for independent, identically distributed sequences, i.e., high exceedances appear individually, and θ > 0 for “almost all” cases of interest. The estimation of the extremal index through the use of the Generalized Jackknife methodology, possibly together with the use of subsampling techniques, is performed. Case studies in the fields of environment and finance will illustrate the performance of the new extremal index estimator comparatively to the classical one.

The upcrossings index and the extremal index

For stationary sequences X = {Xn}n≥1 we relate τ, the limiting mean number of exceedances of high levels un by X1,…,Xn, and ν, the limiting mean number of upcrossings of the same level, through the expression θ = (ν/τ)η, where θ is the extremal index of X and η is a new parameter here called the upcrossings index. The upcrossings index is a measure of the clustering of upcrossings of u by variables in X, and the above relation extends the known relation θ = ν/τ, which holds under the mild-oscillation local restriction D″(u) on X. We present a new family of local mixing conditions D̃(k)(u) under which we prove that (a) the intensity of the limiting point process of upcrossings and η can both be computed from the k-variate distributions of X; and (b) the cluster size distributions for the exceedances are asymptotically equivalent to those for the lengths of one run of exceedances or the lengths of several consecutive runs which are separated by at most k − 2 nonexceedances and, except for the last one, each contain at most k − 2 exceedances.

The upcrossings index and the extremal index

For stationary sequences X = {X n } n≥1 we relate τ, the limiting mean number of exceedances of high levels u n by X 1,…,X n , and ν, the limiting mean number of upcrossings of the same level, through the expression θ = (ν/τ)η, where θ is the extremal index of X and η is a new parameter here called the upcrossings index. The upcrossings index is a measure of the clustering of upcrossings of u by variables in X , and the above relation extends the known relation θ = ν/τ, which holds under the mild-oscillation local restriction D″( u ) on X . We present a new family of local mixing conditions D̃ (k)( u ) under which we prove that (a) the intensity of the limiting point process of upcrossings and η can both be computed from the k-variate distributions of X ; and (b) the cluster size distributions for the exceedances are asymptotically equivalent to those for the lengths of one run of exceedances or the lengths of several consecutive runs which are separated by at most k − 2 nonexceedances and, except for the last one, each contain at most k − 2 exceedances.

The genealogy of a cluster in the multitype voter model

The genealogy of a cluster in the multitype voter model can be defined in terms of a family of dual coalescing random walks. We represent the genealogy of a cluster as a point process in a size-time plane and show that in high dimensions the genealogy of the cluster at the origin has a weak Poisson limit. The limiting point process is the same as for the genealogy of the size-biased Galton-Watson tree. Moreover, our results show that the branching mechanism and the spatial effects of the voter model can be separated on a macroscopic scale. Our proofs are based on a probabilistic construction of the genealogy of the cluster at the origin derived from Harris' graphical representation of the voter model.

Limit distributions for point processes of exceedances of random levels

We give a sufficient condition for the convergence, asn→∞, of the point process of exceedances of the random levels U={un(T)}n≥1 by the random variables of the sequence X={Xn}n≥1, defined by\(S_n [X, U] = \sum\nolimits_{i = 1}^n {1_{\{ X_i > u_n (T)\} } } \delta _{i/n} \). The limiting point process is a doubly stochastic compound Poisson process, whose stochastic intensity measure is regulated by the random variableT. The main result is applied to the study of the asymptotic behaviour of some exceedances point processes of real levels, which can be rewritten as point processes of exceedances of random levels. We consider exceedances by some linear modelsYn=αXn+βT and by chain-dependent sequences, which do not satisfy, in general, the long range dependence condition Δ from Hsinget al. (1988).

On extreme-order statistics and point processes of exceedances in multivariate stationary Gaussian sequences

This paper deals with a weak convergence of extreme-order statistics and point processes of exceedances built on the base of stationary and normal sequences of random vectors. The convolution of normal distribution and a double-exponential-type distribution for the limiting extreme-order statistics, and Cox distribution for the limiting point process of exceedances are found.

On the exceedance point process for a stationary sequence

It is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.

Limiting Point Processes for Rescalings of Coalescing and Annihilating Random Walks on $Z^d$

Let $p(x, y)$ be an arbitrary random walk on $Z^d$. Let $\xi_t$ be the system of coalescing random walks based on $p$, starting with all sites occupied, and let $\eta_t$ be the corresponding system of annihilating random walks. The spatial rescalings $P(0 \in \xi_t)^{1/d}\xi_t$ for $t \geqq 0$ form a tight family of point processes on $R^d$. Any limiting point process as $t \rightarrow\infty$ has Lesbegue measure as its intensity, and has no multiple points. When $p$ is simple random walk on $Z^d$ these rescalings converge in distribution, to the simple Poisson point process for $d \geq 2$, and to a non-Poisson limit for $d = 1$. For a large class of $p$, we prove that $P(0 \in \eta_t)/P(0 \in \xi_t) \rightarrow 1/2$ as $t \rightarrow\infty$. A generalization of this result, proved for nearest neighbor random walks on $Z^1$, and for all multidimensional $p$, implies that the limiting point process for rescalings $P(0 \in \xi_t)^{1/d}\eta_t$ of the system of annihilating random walks is the one half thinning of the limiting point process for the corresponding coalescing system.

Limiting point processes in the branching random walk

Take the nth generation of a supercritical branching random walk (a spatially homogeneous branching process) as a process of cluster centres and take independent copies of some simple point process Y as the clusters. Let the resulting point process be Yn. For a given sequence of real numbers {xn} let Yn be centred on xn. Under certain conditions, when an appropriate scale change is made, the resulting point process converges in distribution to a non-trivial limit.

On a class of random motions of point processes involving interaction

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.

On a class of random motions of point processes involving interaction

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.