Joint limit distributions of exceedances point processes and partial sums of gaussian vector sequence

In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence (centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.     

Weak convergence of the tail empirical process for dependent sequences

This paper proves weak convergence in D$D$ of the tail empirical process–the renormalized extreme tail of the empirical process–for a large class of stationary sequences. The conditions needed for convergence are (i) moment restrictions on the amount of clustering of extremes, (ii) restrictions on long range dependence (absolute regularity or strong mixing), and (iii) convergence of the covariance function. We further show how the limit process is changed if exceedances of a nonrandom level are replaced by exceedances of a high quantile of the observations. Weak convergence of the tail empirical process is one key to asymptotics for extreme value statistics and its wide range of applications, from geoscience to finance.     

一类Gauss序列极值的极限律

本文讨论一类非平稳Gauss序列的极值．利用点过程收敛定理得到多水平超过的点过程的收敛性,同时得到在不相交区间上最大值的联合渐近分布,第k个最大值的渐近分布以及前r个极值的联合渐近状态．     

On the exceedance point process for a stationary sequence

Summary 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.This research has been supported by the Air Force Office of Scientific Research Grant No. F 49620 85 C 0144 and the Katholieke Universiteit Leuven     

一类拟平稳序列极值的极限律

Ｋ．Ｆ．Ｔｕｒｋｍａｎ讨论了一类拟平稳序列最大值的渐近分布。本文利用点过程收全党一理得到水平超出点过程的收敛定理和第ｒ个最大值的渐近分布及前ｒ个最大值的联合渐近分布。     

Quadratically and superlinearly convergent algorithms for the solution of inequality constrained minimization problems

In this paper, some Newton and quasi-Newton algorithms for the solution of inequality constrained minimization problems are considered. All the algorithms described produce sequences {x _k } convergingq-superlinearly to the solution. Furthermore, under mild assumptions, aq-quadratic convergence rate inx is also attained. Other features of these algorithms are that only the solution of linear systems of equations is required at each iteration and that the strict complementarity assumption is never invoked. First, the superlinear or quadratic convergence rate of a Newton-like algorithm is proved. Then, a simpler version of this algorithm is studied, and it is shown that it is superlinearly convergent. Finally, quasi-Newton versions of the previous algorithms are considered and, provided the sequence defined by the algorithms converges, a characterization of superlinear convergence extending the result of Boggs, Tolle, and Wang is given.This research was supported by the National Research Program Metodi di Ottimizzazione per la Decisioni, MURST, Roma, Italy.     

Set partitions and moments of random variables

It is well known that the sequence of Bell numbers (Bn)_n?0 (Bn being the number of partitions of the set [n]) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence (Bn+1)_n?0 is the sequence of moments of 1+τ. In this paper, we generalize these results by showing that both and (where is the number of m-partitions of [n], as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when is replaced by the number of ordered m-partitions of [n]. In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes.     

Statistical estimation for some dividend problems under the compound Poisson risk model

In this paper, we consider some dividend problems in the classical compound Poisson risk model under a constant barrier dividend strategy. Suppose that the Poisson intensity for the claim number process and the distribution for the individual claim sizes are both unknown. We use the COS method to study the statistical estimation for the expected present value of dividend payments before ruin and the expected discounted penalty function. The convergence rates under large sample setting are derived. Some simulation results are also given to show effectiveness of the estimators under finite sample setting.     

The compound Poisson distribution and return times in dynamical systems

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 limiting distribution is a compound Poissonian distribution. We also derive error terms for the convergence to the limiting distribution. We also prove a very general theorem that can be used to establish compound Poisson distributions in many other settings.     

Statistical Cluster Points of Sequences in Finite Dimensional Spaces

In this paper we study the set of statistical cluster points of sequences in m-dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m-dimensional spaces too. We also define a notion of -statistical convergence. A sequence xis -statistically convergent to a set Cif Cis a minimal closed set such that for every > 0 the set has density zero. It is shown that every statistically bounded sequence is -statistically convergent. Moreover if a sequence is -statistically convergent then the limit set is a set of statistical cluster points.     

On a Berry-Esseen theorem for compound Poisson processes

The well-known Berry-Esseen theorem concerning the rate of convergence to a stable law for a sum of independent identically distributed (i.i.d.) random variables is adapted to the case of a compound Poisson process, considered in the collective risk theory. As a consequence the rate of convergence of the Edgeworth expansion to the compound Poisson distribution is examined for all positive values of the time variable, in both cases where the moments of the claim distribution converge or diverge. As a by product the results obtained by T. Höglund [1] concerning the sum of a fixed number (n) of i.i.d. random variables are presented in an alternative manner. His theorems concerning the limiting behaviour for n → ∞ can be transformed slightly in order to make them hold for all n. It is explained how the result on the estimation of the rate of convergence in a limit theorem with a stable law fits with the results obtained by K.I. Satyabaldina [2].     

Series representation for semistable laws and their domains of semistable attraction

If the centered and normalized partial sums of an i.i.d. sequence of random variables converge in distribution to a nondegenerate limit then we say that this sequence belongs to the domain of attraction of the necessarily stable limit. If we consider only the partial sums which terminate atk _n wherek _n+1 ck _n then the sequence belongs to the domain of semistable attraction of the necessarily semistable limit. In this paper, we consider the case where the limiting distribution is nonnormal. We obtain a series representation for the partial sums which converges almost surely. This representation is based on the order statistics, and utilizes the Poisson process. Almost sure convergence is a useful technical device, as we illustrate with a number of applications.This research was supported by a research scholarship from the Deutsche Forschungsgemeinschaft (DFG).     

Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series

We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p _n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavy-tailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.     

Extreme value theory for continuous parameter stationary processes

Summary In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences.The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite mean(x), it is found that the classical criteria for domains of attraction apply when(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for which(x) may or may not be finite).A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr ^th largest local maxima.This work was supported by the Office of Naval Research under Contract N00014-75-C-0809, and in part by the Danish natural Science research Council     

Epigraphical and Uniform Convergence of Convex Functions

We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of .

     

Analysis of multi-level queueing systems with servers breakdown by using recursive solution technique

In this paper we construct a multi-level queueing model that alternates between three modes of an operation system. The service times have followed an Erlang type K   distribution with parameter μ$\mu$. Customers arrive in batches according to a time-homogeneous compound Poisson process with mean rate λ$\lambda$ for the batches. Our aim is to give a recursive scheme for the solution of the steady state equations. Next we derive some important measures of performance which may affect the efficiency of the system under consideration such as the expected waiting time per customer, the expected number of customers who arrive to a full system. The expected number of customers will also be calculated. Finally, we can also calculate the efficiency measures of the system by using the recursive results through an example.     

Über Konvergenz von Spektralfolgen der stabilen Homotopietheorie

We discuss the problem of convergence of spectral sequences that arise from a filtration of a spectrum in Boardman's stable homotopy category by applying a generalized homology, homotopy or cohomology theory. The criteria we get give e.g. the convergence of the Adams spectral sequence for a generalized homology theory in certain cases (using similar methods this equestion has been considered independently by J. F. Adams in his forthcoming Chicago lecture notes), and some results on the Adams cohomology spectral sequence including the well-known convergence properties in case of singular cohomology with Z_p-coefficients and complex cobordism.     

Convergence Properties of the Successive Extrapolated Relaxation (S.E.R.) Method

Extrapolation methods to accelerate convergence of a sequenceof iterates are investigated. A transformation formula derivedfrom the related deterministic sequence is modified so thatit may be used for the stochastic sequences. The S.E.R. method,which is related to Aitken's ² process, is discussed. For linearlyconvergent sequences it is shown that S.E.R. not only will convergeif the original sequence converges, but will converge to thesame limit. An analysis of the bounds for the convergence andthe perturbations is made for Aitken's ² process, S.E.R. andS.E.O.R. The method is applicable to convergent and locallyconvergent vector sequences.