Marcinkiewicz–Zygmund Type Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

Etemadi (in Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119–122, 1981) proved that the Kolmogorov strong law of large numbers holds for pairwise independent identically distributed (pairwise i.i.d.) random variables. However, it is not known yet whether the Marcinkiewicz–Zygmund strong law of large numbers holds for pairwise i.i.d. random variables. In this paper, we obtain the Marcinkiewicz–Zygmund type strong law of large numbers for pairwise i.i.d. random variables {X _n ,n≥1} under the moment condition E|X ₁| ^p (loglog|X ₁|)^2(p?1)<∞, where 1<p<2.     

Weak ergodicity of stationary pairwise independent processes

It is proven that a stationary process of pairwise independent random variables with values in a separable metric space is weakly ergodic, i.e. each random variable is independent of the system of invariant sets of the process. An example shows that a process of identically distributed pairwise independent random variables is in general, however, not weakly ergodic.

     

Robust numerical integration and pairwise independent random variables

A numerical integration method by means of random samples is called robust, if the variance of its error is as small as that of the i.i.d.-sampling for any integrand. To reduce the randomness of robust numerical integration, we use pairwise random samples instead of i.i.d. samples. Among others, we recommend the discrete random Weyl sampling for the quick generation of pairwise independent samples.     

On the laws of large numbers for nonnegative random variables

Strong laws of large numbers concerning nonnegative random variables are obtained and then they are utilized to establish stability results, among other things, for sums of pairwise independent random variables and the range of random walks.     

The von Bahr–Esseen moment inequality for pairwise independent random variables and applications

In the paper, we generalize the von Bahr–Esseen moment inequality from independent random variables to pairwise independent random variables. As the applications, the moment convergence, the complete convergence and the strong law of large numbers are established for pairwise independent random variables.     

两两独立序列加权和的L~r收敛性

讨论了两两独立随机变量列加权和在满足r(1≤r<2)阶Ces`aro一致可积条件下的Lr收敛性,获得了与独立情形一致的结果.用相似的方法,对于其它相依或混合序列(如两两NQD列,φ-混合序列,ρ-混合序列)也有相同的结果.     

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.     

两两独立同分布序列Cesàro强大数定律的收敛速度

设{X,Xn,n≥0}是两两独立同分布的随机变量序列,11．为了证明这一结论而获得到的两两负相关随机变量序列的Cesaro强大数定律收敛速度的结果本身也是有意义的．此结果对于同分布的两两NQD序列也是对的．     

重尾情形下NA随机变量的随机加权和

In 2003, Tang Qihe et al. obtained a simple asymptotic formula for independent identically distributed （i.i.d.） random variables with heavy tails. In this paper, under certain moment conditions, we establish a formula as the same as Tang’s, when random variables are negatively associated （NA）.     

关于两两NQD随机序列的一个极限定理

设为两两NQD随机序列, 且, 是一列严格单调递增的凸序列. 本文将 Feller (1946)关于独立同分布期望不存在随机序列的极限定理推广到两两NQD随机 序列的情形.     

An elementary proof of the strong law of large numbers

Summary In the following note we present a proof for the strong law of large numbers which is not only elementary, in the sense that it does not use Kolmogorov's inequality, but it is also more applicable because we only require the random variables to be pairwise independent. An extension to separable Banach space-valuedr-dimensional arrays of random vectors is also discussed. For the weak law of large numbers concerning pairwise independent random variables, which follows from our result, see Theorem 5.2.2 in Chung [1].     

Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.     

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.     

The almost equivalence of pairwise and mutual independence and the duality with exchangeability

For a large collection of random variables in an ideal setting, pairwise independence is shown to be almost equivalent to mutual independence. An asymptotic interpretation of this fact shows the equivalence of asymptotic pairwise independence and asymptotic mutual independence for a triangular array (or a sequence) of random variables. Similar equivalence is also presented for uncorrelatedness and orthogonality as well as for the constancy of joint moment functions and exchangeability. General unification of multiplicative properties for random variables are obtained. The duality between independence and exchangeability is established through the random variables and sample functions in a process. Implications in other areas are also discussed, which include a justification for the use of mutually independent random variables derived from sequential draws where the underlying population only satisfies a version of weak dependence. Macroscopic stability of some mass phenomena in economics is also characterized via almost mutual independence. It is also pointed out that the unit interval can be used to index random variables in the ideal setting, provided that it is endowed together with some sample space a suitable larger measure structure. Received: 16 April 1997 / Revised version: 18 May 1998     

Simple measure of similarity for the market graph construction

A simple measure of similarity for the construction of the market graph is proposed. The measure is based on the probability of the coincidence of the signs of the stock returns. This measure is robust, has a simple interpretation, is easy to calculate and can be used as measure of similarity between any number of random variables. For the case of pairwise similarity the connection of this measure with the sign correlation of Fechner is noted. The properties of the proposed measure of pairwise similarity in comparison with the classic Pearson correlation are studied. The simple measure of pairwise similarity is applied (in parallel with the classic correlation) for the study of Russian and Swedish market graphs. The new measure of similarity for more than two random variables is introduced and applied to the additional deeper analysis of Russian and Swedish markets. Some interesting phenomena for the cliques and independent sets of the obtained market graphs are observed.     

THE ERDOS-RENYI LAWS OF LARGE NUMBERS FOR NON-IDENTICALLY DISTRIBUTED RANDOM VARIABLES

The Erdos-Renyi law of large numbers (1970) is the first important result forasymptotic behaviours of increments of partial sams of a sequence of random variableswith apan [ClogN]. Some generalizations have been done sinoe then, such as conver-gence rate of the limit, some results when order of span being either higher or lowerthan log N. But all these results are only obtained in the case of i. i. d. random variables.This paper aims at the generalization of these results to the ease when random variablesare independent, but not necessarily identically distributed. To this end Chernoff Theoremis generalized to the corlesponding case at first.     

Convergence in law of random sums with non-random centering

We extend results obtained in Kruglov,⁽⁷⁾ and Finkelstein and Tucker⁽³⁾ to obtain necessary and sufficient conditions for convergence in law of random sums of non-identically distributed independent random variables under non-random centering. Thei.i.d. case is also considered for random variables attracted to a stable law. Necessary and sufficient conditions for convergence in law of these random variables under non-random centering, and in some cases, under non-random norming, are also obtained. The distribution functions for the limit laws are determined as well, generalizing results of Robbins.⁽¹⁰⁾ Supported in part by The State University of New York and United States Information Agency Grant No. IA AEMP69193692.     

关于大偏差概率的一个界

研究得到了关于随机和S(t)=∑N(t)i=1Xi,t≥0大偏差的幂的一个界,其中(N(t))t≥0是一族非负整值随机变量,(Xn)n∈N是独立同分布的随机变量,其共同的分布函数是F与(N(t))t≥0独立.本结论是在假设分布函数F的右尾属于ERV族的情况下得到的.     

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Letbe a sequence of real-valued random variables and be other random variables which are independent of the form sequence. Suppose that are pairwise generalized negatively orthant dependent with heavy tails under the condition that are independent or associated, some asymptotic formulas are established.     

On Kolmogorov SLLN Under Rearrangements for “Orthogonal” Random Variables in a B-Space

It is well-known that the Kolmogorov SLLN (non-i.i.d. case) fails for pair-wise independent random variables. However, as shown in the paper it can be saved even for orthogonal random variables if one allows permutations. We prove it in the setup of Banach space valued random variables.