The distribution of partially exchangeable random variables

In this article, we derive the distribution of partially exchangeable binary random variables, generalizing the distribution of exchangeable binary random variables and hence the binomial distribution. The distribution can be viewed as a mixture of Markov chains. We introduce rectangular complete monotonicity and show that partial exchangebility can be characterized by rectangular complete monotonicity. The distribution aided with rectangular complete monotonicity can be used to analyze serially correlated data common in many areas of science.     

A De Finetti-type theorem for nonexchangeable finite-valued random variables

We introduce the concept of a random assignment process (roughly speaking, such a process assigns, according to some weight function, labels to the points of an arbitrary set), and we impose conditions on the weight function ensuring that a De Finetti-type theorem is satisfied. In particular, this provides a wide class of finite-valued random variables which, despite they are neither exchangeable nor identically distributed, verify a De Finetti-type theorem (i.e., they are conditionally independent).     

Exchangeable and partially exchangeable random partitions

Summary Call a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn ₁,...,n _k, the probability that the partition breaks the firstn ₁+...+n_k integers intok particular classes, of sizesn ₁,...,n_k in order of their first elements, has the same valuep(n ₁,...,n_k) for every possible choice of classes subject to the sizes constraint. A random partition is exchangeable iff it is partially exchangeable for a symmetric functionp(n ₁,...n_k). A representation is given for partially exchangeable random partitions which provides a useful variation of Kingman's representation in the exchangeable case. Results are illustrated by the two-parameter generalization of Ewens' partition structure.Research supported by N.S.F. Grants MCS91-07531 and DMS-9404345     

Representations by uncorrelated random variables

All multivariate random variables with finite variances are univariate functions of uncorrelated random variables and if the multivariate distribution is absolutely continuous then these univariate functions are piecewise linear. They can be independent of the correlations in the Gaussian case.     

Frequentistic approximations to Bayesian prevision of exchangeable random elements

Posterior and predictive distributions for m future trials, given the first n elements of an infinite exchangeable sequence ${\tilde{\xi }}_1,{\tilde{\xi }}_2,\dots$, are considered in a nonparametric Bayesian setting. The former distribution is compared to the unit mass at the empirical distribution ${\tilde{\mathfrak{e}}}_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{{\tilde{\xi }}_i}$ of the n past observations, while the latter is compared to the m-fold product ${\tilde{\mathfrak{e}}}_n^m$. Comparisons are made by means of distinguished probability distances inducing topologies that are equivalent to (or finer than) the topology of weak convergence of probability measures. After stating almost sure convergence to zero of these distances as n goes to infinity, the paper focuses on the analysis of the rate of approach to zero, so providing a quantitative evaluation of the approximation of posterior and predictive distributions through their frequentistic counterparts ${\delta }_{{\tilde{\mathfrak{e}}}_n}$ and ${\tilde{\mathfrak{e}}}_n^m$, respectively. Characteristic features of the present work, with respect to more common literature on Bayesian consistency, are: first, comparisons are made between entities which depend on the n past observation only; second, the approximations are studied under the actual (exchangeable) law of the ${\tilde{\xi }}_n$'s, and not under hypothetical product laws ${\mathfrak{p}}_0^{\infty }$, as ${\mathfrak{p}}_0$ varies among the admissible determinations of a random probability measure.     

Order statistics from mixed exchangeable random variables

Two different exchangeable samples are considered and these two samples are assumed to be independent of each other. From these two samples a new sample is combined and treated as a single set of observations. The distribution of a single order statistic and the joint distribution of two order statistics for a new mixed sample are derived and expressed in terms of joint distribution functions. As a special case the distribution of a single order statistic and the joint distribution of two nonadjacent order statistics from exchangeable random variables are obtained. The results presented in this paper allows widespread applications in modelling of various lifetime data, biomedical sciences, reliability and survival analysis, actuarial sciences etc., where the assumption of independence of data cannot be accepted and the exchangeability is a more realistic assumption.     

Circular law for random matrices with exchangeable entries

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the uniform law on the unit disc. This is an instance of the universality phenomenon known as the circular law, for a model of random matrices with dependent entries, rows, and columns. It is also a non‐Hermitian counterpart of a result of Chatterjee on the semi‐circular law for random Hermitian matrices with exchangeable entries. The proof relies in particular on a reduction to a simpler model given by a random shuffle of a rigid deterministic matrix, on hermitization, and also on combinatorial concentration of measure and combinatorial Central Limit Theorem. A crucial step is a polynomial bound on the smallest singular value of exchangeable random matrices, which may be of independent interest. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 454–479, 2016     

Sequences and arrays involving free variables

Abstact: Sequences in free variables are introduced and used to construct arrays in free variables which are suitable for circulant matrices. Most of the arrays found are maximal in the number of free variables. Applications include many new Goethals‐Seidel type arrays and complex orthogonal designs. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 17–27, 2001     

Some partial orderings of exchangeable random variables by positive dependence

Some partial orderings of positively dependent exchangeable random variables are introduced. The interrelations among them, the inequalities which follow from them and two models which yield such partial orderings are then discussed. Particular examples include ordering multivariate normal, t, χ², Cauchy, exponential, binomial, Poisson, gamma and Farlie-Gumbel-Morgenstern random vectors. Applications to genetic selection and choice of sampling procedures are given.     

A maximal inequality for partial sums of finite exchangeable sequences of random variables

Let be a finite exchangeable sequence of Banach space valued random variables, i.e., a sequence such that all joint distributions are invariant under permutations of the variables. We prove that there is an absolute constant such that if , then

for all . This generalizes an inequality of Montgomery-Smith and Lata{\l}a for independent and identically distributed random variables. Our maximal inequality is apparently new even if is an infinite exchangeable sequence of random variables. As a corollary of our result, we obtain a comparison inequality for tail probabilities of sums of arbitrary random variables over random subsets of the indices.

     

NA随机变量阵列的完全矩收敛性

利用NA随机变量的矩不等式和截尾方法,研究了NA随机变量阵列的完全矩收敛性,给出了证明NA随机变量阵列完全矩收敛性的一些充分条件.所得结果推广了已有文献关于NA随机变量的相应结果.     

Complete moment convergence for arrays of rowwise NSD random variables

In this paper, the complete convergence and complete moment convergence for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables are investigated. Some sufficient conditions to prove the complete convergence and the complete moment convergence are presented. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

Weak laws of large numbers for arrays of dependent random variables

In this paper, we establish some weak laws of large numbers for arrays of dependent random variables satisfying the conditions of a kind of uniform integrability. Our results extend and improve the corresponding ones.     

ND阵列加权乘积和的完全收敛性

设（Xni：1≤i≤n,n≥1）为行间ND阵列,g（x）是R^＋上指数为α的正则变化函数,{αni：1≤i≤n,n≥1}为满足条件max1≤i≤n｜ani｜=0（（g（n））^-1）的实数阵列．本文采用截尾的方法,得到了使ND随机变量阵列加权乘积和完全收敛的条件,并推广了以前学者的结论．     

A central limit theorem for exchangeable random variables on a locally compact abelian group

A central limit theorem is given for uniformly infinitesimal triangular arrays of random variables in which the random variables in each row are exchangeable and take values in a locally compact second countable abelian group. The limiting distribution in the theorem is Gaussian.