Almost Sure Convergence of Weighted Sums for Negatively Associated Random Variables

In this article, let {X₁, …, X_n} be a sequence of negatively associated random variables and {a_ni, 1 ? i ? n, n ? 1} be a triangular array of constants. Several almost sure convergence theorems for the weighted sums ∑ⁿ_{i = 1}a_niX_i are established.     

Complete Convergence for Weighted Sums and Arrays of Rowwise Extended Negatively Dependent Random Variables

In this article, we study the complete convergence for weighted sums of extended negatively dependent random variables and row sums of arrays of rowwise extended negatively dependent random variables. We apply two methods to prove the results: the first of is based on exponential bounds and second is based on the generalization of the classical moment inequality for extended negatively dependent random variables.     

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

In this article, we establish a new complete convergence theorem for weighted sums of negatively dependent random variables. As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008 Jing, B.Y., Liang, H.Y. (2008). Strong limit theorems for weighted sums of negatively associated random variables. J. Theor. Probab. 21:890–909.[Crossref], [Web of Science ®] , [Google Scholar]), Sung (2012 Sung, S.H. (2012). Complete convergence for weighted sums of negatively dependent random variables. Stat. Pap. 53:73–82.[Crossref], [Web of Science ®] , [Google Scholar]), and Wu (2010) can be obtained.     

Complete moment convergence of weighted sums for arrays of negatively dependent random variables and its applications

For testing goodness-of-fit in a k cell multinomial distribution having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio statistic, G² is not satisfactory. A new adjustment to G² is determined on the basis of analytical investigation in terms of asymptotic bias and variance of the adjusted G² A Monte Carlo simulation is performed for several one-way tables to assess the adjustment of G² in order to obtain a closer approximation to the nomial level of significance.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

On Complete Convergence in Marcinkiewicz-Zygmund Type SLLN for END Random Variables and Its Applications

Abstract

In this paper, the complete convergence for maximal weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence and some applications to a nonparametric model are provided. The results obtained in the paper generalize and improve the corresponding ones of Wang et al. (2014 Wang, X. J., X. Deng, L. L. Zheng, and S. H. Hu. 2014. Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications. A Journal of Theoretical and Applied Statistics 48(4):834–50. [Google Scholar]b) and Shen, Xue, and Wang (2017 Shen, A., M. Xue, and W. Wang. 2017. Complete convergence for weighted sums of extended negatively dependent random variables. Communications in Statistics – Theory and Methods 46(3):1433–44.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).     

On Complete Convergence for Arrays of Dependent Random Variables

A complete convergence theorem for an array of rowwise independent random variables was established by Sung et al. (2005 Sung , S. H. , Volodin , A. I. , Hu , T.-C. ( 2005 ). More on complete convergence for arrays . Statist. Probab. Lett. 71 : 303 – 311 .[Crossref], [Web of Science ®] , [Google Scholar]). This result has been generalized and extended by Kruglov et al. (2006 Kruglov , V. M. , Volodin , A. I. , Hu , T.-C. ( 2006 ). On complete convergence for arrays . Statist. Probab. Lett. 76 : 1631 – 1640 .[Crossref], [Web of Science ®] , [Google Scholar]) and Chen et al. (2007 Chen , P. , Hu , T.-C. , Liu , X. , Volodin , A. ( 2007 ). On complete convergence for arrays of rowwise negatively associated random variables . Theor. Probab. Appl. 52 : 393 – 397 . [Google Scholar]). In this article, we extend the results of Sung et al. (2005 Sung , S. H. , Volodin , A. I. , Hu , T.-C. ( 2005 ). More on complete convergence for arrays . Statist. Probab. Lett. 71 : 303 – 311 .[Crossref], [Web of Science ®] , [Google Scholar]), Kruglov et al. (2006 Kruglov , V. M. , Volodin , A. I. , Hu , T.-C. ( 2006 ). On complete convergence for arrays . Statist. Probab. Lett. 76 : 1631 – 1640 .[Crossref], [Web of Science ®] , [Google Scholar]), and Chen et al. (2007 Chen , P. , Hu , T.-C. , Liu , X. , Volodin , A. ( 2007 ). On complete convergence for arrays of rowwise negatively associated random variables . Theor. Probab. Appl. 52 : 393 – 397 . [Google Scholar]) to an array of dependent random variables satisfying Hoffmann-Jørgensen type inequalities.     

Complete convergence for arrays of row-wise ND random variables under sub-linear expectations

In this paper, complete convergence for arrays of row-wise ND random variables under sub-linear expectations is studied. As applications, the complete convergence theorems of weighted sums for negatively dependent random variables have been generalized to the sub-linear expectation space context. We extend some complete convergence theorems from the traditional probability space to the sub-linear expectation space and our results generalize corresponding results obtained by Ko.     

Equivalent Conditions of Complete Moment Convergence of Weighted Sums for ϕ-Mixing Sequence of Random Variables

In this article, the complete moment convergence of weighted sums for ?-mixing sequence of random variables is investigated. By applying moment inequality and truncation methods, the equivalent conditions of complete moment convergence of weighted sums for ?-mixing sequence of random variables are established. These results promote and improve the corresponding results obtained by Li et al. (1995 Li, D.L., Rao, M.B., Jiang, T.F., Wang, X.C. (1995). Complete convergence and almost sure convergence of weighted sums of random variables. J. Theoret. Probab. 8:49–76.[Crossref], [Web of Science ®] , [Google Scholar]) and Gut (1993 Gut, A. (1993). Complete convergence and Cesàro summation for i.i.d. random variables. Probab. Theory Related Fields 97:169–178.[Crossref], [Web of Science ®] , [Google Scholar]) from i.i.d. to ?-mixing setting. Moreover, we obtain the complete moment convergence of moving average processes based on ?-mixing random variables, which extends the result of Kim et al. (2008 Kim, T.S., Ko, M.H. (2008). Complete moment convergence of moving average processes under dependence assumptions. Statist. Probab. Lett. 78:839–846.[Crossref], [Web of Science ®] , [Google Scholar]) in the sense that it does not require a specific mixing rate.     

On Complete Convergence for an Extended Negatively Dependent Sequence

In this article, the Rosenthal-type maximal inequality for extended negatively dependent (END) sequence is provided. By using the Rosenthal type inequality, we present some results of complete convergence for weighted sums of END random variables under mild conditions.     

On the convergence rate for arrays of row-wise NOD random variables

Abstract

In this article, the complete convergence results of weighted sums for arrays of rowwise negatively orthant dependent (NOD) random variables are investigated. Some sufficient conditions for complete convergence for arrays of rowwise NOD random variables are presented without assumption of identical distribution.     

Complete moment convergence for weighted sums of extended negatively dependent random variables

Complete moment convergence for weighted sums of sequence of extended negatively dependent (END) random variables is discussed. Some new sufficient and necessary conditions of complete moment convergence for weighted sums of END random variables are obtained, which improve and extend some well-known results in the literature.     

Convergence and Divergence of Random Series

This paper amplifies Daley's (1981) criteria for absolute convergence of certain random series by providing a sufficient condition which also is necessary if the summands are independent. Conditions for unconditional and conditional convergence are also given. These results are used to obtain a substantially complete picture of the behaviour of random Dirichlet series of a fairly general type. Behaviour of the partial sums of divergent series is discussed, with particular attention to Dirichlet series.     

Convergence for weighted sums of widely orthant dependent random variables

ABSTRACT

In this article, a complete convergence result and a complete moment convergence result are obtained for the weighted sums of widely orthant dependent random variables under mild conditions. As corollaries, the corresponding results are also obtained under the extended negatively orthant dependent setup. In particular, the complete convergence result generalizes and improves the related known works in the literature.     

Complete convergence for weighted sums of extended negatively dependent random variables

In this article, the complete convergence for weighted sums of extended negatively dependent (END, in short) random variables without identical distribution is investigated. In addition, the complete moment convergence for weighted sums of END random variables is also obtained. As an application, the Baum–Katz type result for END random variables is established. The results obtained in the article extend the corresponding ones for independent random variables and some dependent random variables.     

Convergence of weighted sums for sequences of pairwise NQD random variables

ABSTRACT

The authors discuss the convergence for weighted sums of pairwise negatively quadrant dependent (NQD) random variables and obtain some new results which extend and improve the result of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar]. In addition, we relax some restrictions of the conditions in their result. Some new methods are used in this article which differ from that of Bai and Cheng (2000) Bai, Z.D., Cheng, P.E. (2000). Marcinkiewicz strong laws for linear statistics. Stat. Probab. Lett. 46:105–112.[Crossref], [Web of Science ®] , [Google Scholar].     

Complete convergence and complete moment convergence for widely orthant-dependent random variables

In this paper, we first establish the complete convergence for weighted sums of widely orthant-dependent (WOD, in short) random variables by using the Rosenthal type maximal inequality. Based on the complete convergence, we further study the complete moment convergence for weighted sums of arrays of rowwise WOD random variables which is stochastically dominated by a random variable X. The results obtained in the paper generalize the corresponding ones for some dependent random variables.     

On the complete convergence for randomly indexed partial sums and the limit behaviour of a sequence of quantiles

This note contains result on the complete convergence of randomly indexed partial sums of independent non-identically distributed random variables. We use them to investigate the limit behaviour of quantiles of a sample with a random numbers of items.