Convergence results for a normalized triangular array of symmetric random variables

For a triangular array of symmetric random variables (without any integrability condition) we replace the classical assumption of row-wise independence by that of row-wise joint symmetry. Under this weaker assumption we prove some results concerning the convergence in distribution of a suitable sequence of randomly normalized sums to the standard normal distribution. Then we exhibit a class of row-wise independent triangular arrays for which the ordinary sums fail to converge in distribution, while our results enable us to affirm the convergence in distribution of the normalized sums.     

On complete moment convergence of weighted sums for arrays of row-wise negatively associated random variables

Negatively associated (NA) random variables are a more general class of random variables which include a set of independent random variables and have been applied to many practical fields. In this paper, the complete moment convergence of weighted sums for arrays of row-wise NA random variables is investigated. Some sufficient conditions for complete moment convergence of weighted sums for arrays of row-wise NA random variables are established. Moreover, under the weaker conditions, we extend the results of Baek et al. [J. Korean Stat. Soc. 37 (2008), pp. 73–80] and Sung [Abstr. Appl. Anal. 2011 (2011)]. As an application, the complete moment convergence of moving average processes based on an NA random sequence is obtained, which improves the result of Li and Zhang [Stat. Probab. Lett. 70 (2004), pp. 191–197 ].     

The S-Related Dynamic Convex Valuation in the Brownian Motion Setting

Let be an array of row-wise exchangeable random elements in a separable Banach space. Strong laws of large numbers are obtained for under certain moment conditions on the random variables and a condition relating to nonorthogonality. By using reverse martingale techniques, similar results are obtained for triangular arrays of random elements inseparable Banach spaces which are row-wise exchangeable     

Asymptotical behavior of the variance of sums of random variables with random replacements

In this paper, we introduce a scheme of summation of independent random variables with random replacements. We consider a series of double arrays of identically distributed random variables that are row-wise independent, but such that neighboring rows contain a random common part of the repeating terms. By this-scheme we bscribe a model of strongly dependent noise. To investigate the sample mean of this noise, we consider the sum of random variables over the whole double array and its conditional variance with respect to replacements. For columns of the arrays we prove a covariance inequality. As a corollary of it, we demonstrate the law of large numbers for conditional variances. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 124–143. Translated by A. Sudakov.     

Attainable rates of convergence of maxima

Any exponential rate of convergence can be obtained for maxima of i.i.d. random variables, while faster than exponential convergence implies that the variables have an extreme value distribution.     

Hölderian Invariance Principle for Triangular Arrays of Random Variables

In this paper, we extend the Hölderian invariance principle of Lamperti [6] to the case of partial-sum processes based on a triangular array of row-wise independent random variables. As an application, we obtain necessary and sufficient conditions for the almost sure (resp. in probability) weak Hölder convergence of partial-sum processes based on bootstrapped samples.     

独立随机序列最大值的几乎处处极限定理

本文研究了独立随机序列最大值分布的几乎必然收敛性.利用有关协方差的不等式和加权平均,获得独立随机序列最大值的几乎处处极限.将独立同分布随机序列的结论,推广了独立但不同分布的情形.     

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In this article, applying the result of complete convergence for negatively associated (NA) random variables which is obtained by Chen et al.\ucite{14}, the equivalent conditions of complete convergence for weighted sums of arrays of row-wise negatively associated random variables is investigated. As a result, the corresponding results of Liang\ucite{11} is generalized, moreover, the proof procedure is simplified greatly which is different from truncation method of Liang's. Thus, Gut's\ucite{13} result on Ces\`{a}ro summation of i.i.d. random variables is extended.     

Supermodular Functions on Finite Lattices

The supermodular order on multivariate distributions has many applications in financial and actuarial mathematics. In the particular case of finite, discrete distributions, we generalize the order to distributions on finite lattices. In this setting, we focus on the generating cone of supermodular functions because the extreme rays of that cone (modulo the modular functions) can be used as test functions to determine whether two random variables are ordered under the supermodular order. We completely determine the extreme supermodular functions in some special cases.     

On the asymptotic location of high values of a stationary sequence

In this paper, we investigate the limiting distribution of the locations related with high values generated by a strictly stationary sequence of random variables. The main tool for this purpose is the so-called local extremes comparison lemma, which enables us to obtain the convergence in distribution of various functionals related with the location of extreme order statistics, including the location of local maxima and the joint locations of the largest order statistics. Furthermore, results about the joint asymptotic behavior of the location of the first high-level exceedance and the location of the maximum are also discussed.     

Limiting distributions of maxima under triangular schemes

A well-known result in extreme value theory indicates that componentwise taken sample maxima of random vectors are asymptotically independent under weak conditions. However, in important cases this independence is attained at a very slow rate so that the residual dependence structure plays a significant role.In the present article, we deduce limiting distributions of maxima under triangular schemes of random vectors. The residual dependence is expressed by a technical condition imposed on the spectral expansion of the underlying distribution.     

Large-deviation probabilities for maxima of sums of subexponential random variables with application to finite-time ruin probabilities

We establish an asymptotic relation for the large-deviation probabilities of the maxima of sums of subexponential random variables centered by multiples of order statistics of i.i.d.standard uniform random variables.This extends a corresponding result of Korshunov.As an application,we generalize a result of Tang,the uniform asymptotic estimate for the finite-time ruin probability,to the whole strongly subexponential class.     

On families of distributions characterized by certain properties of ordered random variables

In this work, we obtain new characterizations of certain probability distributions by relations with different ordered random variables. Such variables include order statistics, sequential maxima, and records. We consider relations that include not only upper, but also lower record values. The presented ordered objects are based on sequences of independent random variables with a common continuous distribution function. We also investigate equalities in the distribution of sequential maxima exposed by various random shifts. These shifts (one-sided or two-sided) have exponential distributions. Certain theorems and their corollaries present corresponding characterizations of distributions by relations of such a type. In addition, we consider exponentially shifted order statistics such that simple relations among them also characterize certain probability distributions. All of the presented results yield a set of characterizations of various distributions. For particular cases, we present the relations that characterize families of classical exponential and logistic distributions.     

The convergence of the random lines defined by observations of a stochastic process

We deal with independent random variables which are the values of a stochastic process taken at random points in time. So we have random variables depending upon a random parameter. We obtain the conditions providing the weak convergence of random lines defined by sums or maxima or bilinear forms of these random variables for almost all values of the parameter, to one and the same stochastic process. These limit stochastic processes are described. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part II.     

离散指数分布序列的强偏差与强逼近

本文引入离散指数分布概念,建立了关于离散型指数分布序列的强偏差定理和强大数定律.同时,得到离散指数分布序列对连续指数分布序列的强逼近.     

Approximations to the distributions of ordered distance random variables

Summary In the present note we give short proofs of asymptotic theorems for the distributions of extreme and intermediate ordered distance random variables. Moreover, a quick goodness-of-fit test is proposed which is based on a single intermediate ordered distance random variable.     

分布矩阵与随机变量独立性的判定

利用离散型随机变量的联合分布矩阵,得到了离散型随机变量独立性的一种判别方法,并用实例给出了一定的应用。     

Extreme value copula estimation based on block maxima of a multivariate stationary time series

The core of the classical block maxima method consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying series. In asymptotic theory, it is usually postulated that the block maxima are an independent random sample of an extreme value distribution. In practice however, block sizes are finite, so that the extreme value postulate will only hold approximately. A more accurate asymptotic framework is that of a triangular array of block maxima, the block size depending on the size of the underlying sample in such a way that both the block size and the number of blocks within that sample tend to infinity. The copula of the vector of componentwise maxima in a block is assumed to converge to a limit, which, under mild conditions, is then necessarily an extreme value copula. Under this setting and for absolutely regular stationary sequences, the empirical copula of the sample of vectors of block maxima is shown to be a consistent and asymptotically normal estimator for the limiting extreme value copula. Moreover, the empirical copula serves as a basis for rank-based, nonparametric estimation of the Pickands dependence function of the extreme value copula. The results are illustrated by theoretical examples and a Monte Carlo simulation study.     

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This paper considers the asymptotics of randomly weighted sums and their maxima, where the increments {X_i,i\geq1\} is a sequence of independent, identically distributed and real-valued random variables and the weights {\theta_i,i\geq1\} form another sequence of non-negative and independent random variables, and the two sequences of random variables follow some dependence structures. When the common distribution F of the increments belongs to dominant variation class, we obtain some weakly asymptotic estimations for the tail probability of randomly weighted sums and their maxima. In particular, when the F belongs to consistent variation class, some asymptotic formulas is presented. Finally, these results are applied to the asymptotic estimation for the ruin probability.     

New power limits for extremes

For order statistics there is a deceptively simple link between affine and power norming, using exponential transforms. This link does not tell the whole story about limit distributions. The exponential transforms $W=e^{V}$ and $W=-e^{-V}$ yield limit variables which are either positive or negative. Under power norming there exist discrete limit distributions for maxima. The corresponding limit variables assume two values, one of which is zero. All variables with two values, one positive, one zero, are power limits for maxima. They are of different power type if they give different weight to zero, but they all have the same domain, the set of dfs with finite positive upper endpoint and an upper tail which varies slowly. So we see that convergence of types does not hold for power norming. This paper gives a classification of the power limits and their domains for maxima, variables conditioned to be large, and POTs (where power limits may assume three values). Convergence of sample clouds under power norming is studied, and of intermediate upper order statistics. The new power limits do not affect applications. Power norming is a viable alternative to classic extreme value theory. The extra norming constant in the exponent automatically improves the rate of convergence. Hill plots are a good instrument to determine this norming constant. It will be shown how to eliminate the bias of Hill plots and estimate high upper quantiles when the tail does not vary regularly or when convergence is slow.