Complete and complete moment convergence for weighted sums of widely orthant dependent random variables

In this paper, we establish a complete convergence result and a complete moment convergence result for weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results for weighted sums of extended negatively orthant dependent random variables are also obtained, which generalize and improve the related known works in the literature.     

END随机变量阵列加权和的完全收敛性

利用END变量的R0senthal型矩不等式,研究了END随机阵列加权和的完全收敛性,给出了证明完全收敛性的一些充分条件.另外,还给出了证明完全收敛性的一个必要条件.所得结果推广了独立变量和若干相依变量的相应结果.     

Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

Under some conditions of uniform integrability and appropriate conditions, mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables are obtained. Our results extend and improve the results of [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008) 289-300] and [M. Ordóñez Cabrera, A. Volodin, Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305 (2005) 644-658].     

Complete convergence and almost sure convergence of weighted sums of random variables

Letr>1. For eachn1, let {X _nk , –<k<} be a sequence of independent real random variables. We provide some very relaxed conditions which will guarantee for every >0. This result is used to establish some results on complete convergence for weighted sums of independent random variables. The main idea is that we devise an effetive way of combining a certain maximal inequality of Hoffmann-Jørgensen and rates of convergence in the Weak Law of Large Numbers to establish results on complete convergence of weighted sums of independent random variables. New results as well as simple new proofs of known ones illustrate the usefulness of our method in this context. We show further that this approach can be used in the study of almost sure convergence for weighted sums of independent random variables. Convergence rates in the almost sure convergence of some summability methods ofiid random variables are also established.     

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．     

行为NSD随机变量阵列加权和的q阶矩完全收敛性

利用Hoffmann-Jφrgensen型概率不等式和截尾法,获得了行为NSD随机变量阵列加权和的q阶矩完全收敛性的充分条件.利用这些充分条件,不仅推广和深化梁汉营等(2010)和郭明乐等(2014)的结论,而且使他们的证明过程得到了极大地简化.     

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.     

非同分布NA序列的完全收敛性

讨论了非同分布NA序列部分和与随机足标部分和的完全收敛性，推广了于浩在1989年得到的关于独立随机变量序列的一些结果。     

H可积下的相依随机变量和的完全收敛性质

在满足H可积的条件下,利用随机变量的截尾方法,以及相关引理,给出了行内两两NQD序列以及p混合条件的随机组列部分和的完全收敛定理以及强大数定理．     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

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 ].     

Complete moment convergence for weighted sums of widely orthant-dependent random variables and its application in nonparametric regression models

We establish some results on the complete moment convergence for weighted sums of widely orthant-dependent (WOD) random variables, which improve and extend the corresponding results of Y. F. Wu, M. G. Zhai, and J. Y. Peng [J. Math. Inequal., 2019, 13(1): 251–260]. As an application of the main results, we investigate the complete consistency for the estimator in a nonparametric regression model based on WOD errors and provide some simulations to verify our theoretical results.     

Almost sure convergence theorem and strong stability for weighted sums of NSD random variables

In this paper, Kolmogorov-type inequality for negatively superadditive dependent (NSD) random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extends the corresponding results for independent sequences and negatively associated (NA) sequences. In addition, the strong stability for weighted sums of NSD random variables is studied.     

Complete moment convergence of pairwise NQD random variables

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.     

NA序列部分和的矩完全收敛性

讨论了NA序列部分和的矩完全收敛性,在一定条件下获得了NA序列矩完全收敛的充要条件,显示了矩完全收敛和矩条件之间的关系,将独立同分布随机变量序列矩完全收敛的结果推广到NA序列,得到了与独立随机变量序列情形类似的结果.     

On the rate of complete convergence for weighted sums of NSD random variables and an application

In this paper, the complete convergence is established for the weighted sums of negatively superadditive-dependent random variables. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for the random weighted average is also achieved, and a simulation study is done for the asymptotic behaviour of random weighting estimator.     

WOD随机变量加权和的完全收敛性

宽象限相依变量(简称WOD变量)是一类包含独立变量,负相协变量(简称NA变量),负象限相依变量(简称NOD变量)和推广的负象限相依变量(简称END变量)在内的非常广泛的相依变量.本文利用WOD变量的Rosenthal型矩不等式和随机变量的截尾技术,在一般的条件下建立了WOD变量加权和的完全收敛性.所得结果推广了若干相依变量的相应结果.     

加权AANA随机变量和的一致强收敛速度

为了完善 AANA 序列的极限理论,利用三级数定理、Borel-Cantelli 引理及一些概率不等式,研究了AANA 随机变量序列的函数加权和。在一定的条件下,得到了其一致强收敛速度为n？13 log n,推广了关于NA随机变量序列的相应结果。