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.     

The asymptotic distribution of the maxima of a Gaussian random field on a lattice

Under mild conditions on the covariance function of a stationary Gaussian process, the maxima behaves asymptotically the same as the maxima of independent, identically distributed Gaussian random variables. In order to achieve extremal clustering, Hsing et al. (Ann Appl Probab 6:671–686, 1996) considered a triangular array of Gaussian sequences in which the correlation between “neighboring” observations approaches 1 at a certain rate. Using analogues of the conditions of Hsing et al., which allows for strong local dependence among variables but asymptotic independence, it is possible to show that two-dimensional Gaussian random fields also exhibit extremal clustering in the limit. A closed form expression for the extremal index governing the clustering will be provided. The results apply to Gaussian random fields in which the spatial domain is rescaled.     

The Matsumoto-Yor property and the structure of the Wishart distribution

This paper establishes a link between a generalized matrix Matsumoto-Yor (MY) property and the Wishart distribution. This link highlights certain conditional independence properties within blocks of the Wishart and leads to a new characterization of the Wishart distribution similar to the one recently obtained by Geiger and Heckerman but involving independences for only three pairs of block partitionings of the random matrix.In the process, we obtain two other main results. The first one is an extension of the MY independence property to random matrices of different dimensions. The second result is its converse. It extends previous characterizations of the matrix generalized inverse Gaussian and Wishart seen as a couple of distributions.We present two proofs for the generalized MY property. The first proof relies on a new version of Herz's identity for Bessel functions of matrix arguments. The second proof uses a representation of the MY property through the structure of the Wishart.     

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.     

Asymptotic properties of random matrices and pseudomatrices

We study the asymptotics of sums of matricially free random variables, called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called ‘matricially free Gaussian operators’. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are ‘asymptotically matricially free’ whereas the corresponding symmetric random blocks are ‘asymptotically symmetrically matricially free’, where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, block-lower-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.     

Central Limit Theorem for Linear Processes with Infinite Variance

This paper addresses the following classical question: Given a sequence of identically distributed random variables in the domain of attraction of a normal law, does the associated linear process satisfy the central limit theorem? We study the question for several classes of dependent random variables. For independent and identically distributed random variables we show that the central limit theorem for the linear process is equivalent to the fact that the variables are in the domain of attraction of a normal law, answering in this way an open problem in the literature. The study is also motivated by models arising in economic applications where often the innovations have infinite variance, coefficients are not absolutely summable, and the innovations are dependent.     

Extreme Values and the Multivariate Compact Law of the Iterated Logarithm

For a sequence of independent identically distributed Euclidean random vectors, we prove a compact Law of the iterated logarithm when finitely many maximal terms are omitted from the partial sum. With probability one, the limiting cluster set of the appropriately operator normed partial sums is the closed unit Euclidean ball. The result is proved under the hypotheses that the random vectors belong to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The integrability condition characterizes how many maximal terms must be omitted from the partial sum sequence.     

Matricially free random variables

We show that the framework developed by Voiculescu for free random variables can be extended to arrays of random variables whose multiplication imitates matricial multiplication. The associated notion of independence, called matricial freeness, can be viewed as a concept which not only leads to a natural generalization of freeness, but also underlies other fundamental types of noncommutative independence, such as monotone independence and boolean independence. At the same time, the sums of matricially free random variables, called random pseudomatrices, are closely related to random matrices. The main results presented in this paper concern the standard and tracial central limit theorems for random pseudomatrices and the corresponding limit distributions which can be viewed as matricial semicircle laws.     

On the necessity of Statulevičius' condition in limit theorems for large-deviation probabilities

For a sequence of independent identically distributed random variables with zero mean and unit variance, the problem of necessity of Statulevičius' condition in limit theorems for large-deviation probabilities is investigated. St. Petersburg State Technical University. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 293–303, July–September, 1999. Translated by N. N. Amosova     

p—型空间与B值随机元和尾概率的收敛速度

本文讨论了Ｂ值随机元非随机足标与随机足标和尾概率的收敛速度。借助于Ｂ值独立随机元序列满足强大数定律与弱大数定律等价的这一特性，得到了Ｂａｎａｃｈ空间ｐ型性质的刻划，同时将（１，２）中实值独立同分布随机变量和完全收敛性的相应结果推广到Ｂ值独立但不必同分布情形。     

Light tails: all summands are large when the empirical mean is large

It is well known that for a fixed number of independent identically distributed summands with light tail, large values of the sample mean are obtained only when all the summands take large values. This paper explores this property as the number of summands tends to infinity. It provides the order of magnitude of the sample mean for which all summands are in some interval containing this value and it also explores the width of this interval with respect to the distribution of the summands in their upper tail. These results are proved for summands with log-concave or nearly log concave densities. Making use of some extension of the Erdös-Rényi law of large numbers it also explores the forming of aggregates in a sequence of i.i.d. random variables. As a by product the connection is established between large exceedances of the local slope of a random walk on growing bins and the theory of extreme order statistics.     

Fourier-Based Distances and Berry-Esseen Like Inequalities for Smooth Densities

 This paper is devoted to the rate of convergence problem in the central limit theorem for sums of independent identically distributed random variables with regular probability density function. The method we use depends strictly on Fourier based metrics, and yields Berry-Esseen like bounds for the convergence towards both a normal and a stable law in various Sobolev norms. Received 31 May 2001; in revised form 13 November 2001     

An Inductive Proof of the Berry-Esseen Theorem for Character Ratios

Bolthausen used a variation of Stein’s method to give an inductive proof of the Berry-Esseen theorem for sums of independent, identically distributed random variables. We modify this technique to prove a Berry-Esseen theorem for character ratios of a random representation of the symmetric group on transpositions. An analogous result is proved for Jack measure on partitions. Received March 11, 2005     

关于对称矩阵随机变换方差的证明

应用正交变换将对称矩阵对角化,基于随机向量正交变换后独立性的不变性及矩阵迹相关性质,给出一个关于对称矩阵经随机变换后方差的证明,并将该结论推广到更一般情形。     

Polya's Characterization Theorem for Complex Random Variables

A complex random variable can be Gaussian in either the narrow or the wide sense. It is observed that Gaussian random variables in the wide sense do not have the 2-stability property (which is well known for the real case), while in the narrow definition they do possess it. Moreover, it is proved that this property characterizes the class of complex Gaussian random variables in the narrow sense; no other complex random variable enjoys it.     

Circular law, extreme singular values and potential theory

Consider the empirical spectral distribution of complex random n×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.     

一般矩条件下随机变量的收敛性

本文在一般矩条件下研究了同分布的NA随机变量序列和独立同分布的随机变量序列的收敛性,得到了推广形式的Baum-Katz定理和强大数律,这些结果推广了已知的一些文献中相应的结果.     

A note on a dependent risk model with constant interest rate

For a dependent risk model with constant interest rate, in which the claim sizes form a sequence of upper tail asymptotically independent and identically distributed random variables, and their inter-arrival times are another sequence of widely lower orthant dependent and identically distributed random variables, we will give an asymptotically equivalent formula for the finite-time ruin probability. The obtained asymptotics holds uniformly in an arbitrarily finite-time interval.     

Matricial R-transform

We study the addition problem for strongly matricially free random variables which generalize free random variables. Using operators of Toeplitz type, we derive a linearization formula for the matricial R-transform related to the associated convolution. It is a linear combination of Voiculescu?s R-transforms in free probability with coefficients given by internal units of the considered array of subalgebras. This allows us to view this formula as the matricial linearization property of the R-transform. Since strong matricial freeness unifies the main types of noncommutative independence, the matricial R-transform plays the role of a unified noncommutative analog of the logarithm of the Fourier transform for free, boolean, monotone, orthogonal, s-free and c-free independence.     

A Complete Characterization of Multivariate Normal Stable Tweedie Models through a Monge-Ampère Property

Extending normal gamma and normal inverse Gaussian models, multivariate normal stable Tweedie (NST) models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. Within the framework of multivariate exponential families, the NST models are recently classified by their covariance matrices V(m) depending on the mean vector m. In this paper, we prove the characterization of all the NST models through their determinants of V(m), also called generalized variance functions, which are power of only one component of m. This result is established under the NST assumptions of Monge-Ampère property and steepness. It completes the two special cases of NST, namely normal Poisson and normal gamma models. As a matter of fact, it provides explicit solutions of particular Monge-Ampère equations in differential geometry.