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.     

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.     

On complete and complete moment convergence for weighted sums of ANA random variables and applications

In this article, the complete convergence and complete moment convergence for weighted sums of asymptotically negatively associated (ANA, for short) random variables are studied. Several sufficient conditions of the complete convergence and complete moment convergence for weighted sums of ANA random variables are presented. As an application, the complete consistency for the weighted estimator in a nonparametric regression model based on ANA random errors is established by using the complete convergence that we established. We also give a simulation to verify the validity of the theoretical result.     

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.     

Equivalent conditions of the complete convergence for weighted sums of NSD random variables

Abstract

In this paper, we consider the complete convergence for weighted sums of negatively superadditive-dependent (NSD) random variables without assumptions of identical distribution. Some sufficient and necessary conditions to prove the complete convergence for weighted sums of NSD random variables are presented, which extend and improve the corresponding ones of Naderi et al. As an application of the main results, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of NSD random variables is also achieved.     

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.     

The strong consistency of M-estimates in linear models with extended negatively dependent errors

In this paper, we first establish the strong convergence for weighted sums of extended negatively dependent (END) random variables. Based on the strong convergence and Bernstein inequality, we obtain the strong consistency of M-estimates of the regression parameters in a linear model for END random errors under some mild moment conditions. The results generalize and improve the ones obtained in the literature to the case of END random errors.     

Almost sure convergence for weighted sums of WNOD random variables and its applications to non parametric regression models

In this article, some results on almost sure convergence for weighted sums of widely negative orthant dependent (WNOD) random variables are presented. The results obtained in the article generalize and improve the corresponding one of J. Lita Da Silva. [(2015), “Almost sure convergence for weighted sums of extended negatively dependent random variables.” Acta Math. Hungar. 146 (1), 56–70]. As applications, the strong convergence for the estimator of non parametric regression model are established.     

Complete moment convergence and Lr convergence for asymptotically almost negatively associated random variables

In this paper, we investigate the complete moment convergence and L_r convergence for maximal partial sums of asymptotically almost negatively associated random variables under some general conditions. The results obtained in the paper generalize some corresponding ones for negatively associated random variables.     

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.     

Moderate deviations for the random weighted sums of END random variables with consistently varying tails

In this paper, we study moderate deviations for random weighted sums of extended negative dependent (END) random variables, which are consistently-varying tailed and not necessarily identically distributed. When these END random variables are independent of their weights, and the weights are positive random variables with two-sided bounds, the results shows END structure and the dependence between the weights have no effects on the asymptotic behavior of moderate deviations of partial sums and random sums.     

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.     

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.     

General results of complete convergence and complete moment convergence for weighted sums of some class of random variables

ABSTRACT

In the article, the complete convergence and complete moment convergence for weighted sums of sequences of random variables satisfying a maximal Rosenthal type inequality are studied. As an application, the Marcinkiewicz–Zygmund type strong law of large numbers is obtained. Our partial results generalize and improve the corresponding ones of Shen (2013 Shen, A.T. (2013). On strong convergence for weighted sums of a class of random variables. Abstr. Appl. Anal.2013, Article ID 216236: 1–7. [Google Scholar]).     

On the strong convergence of weighted sums of widely dependent random variables

ABSTRACT

For widely dependent random variables, we present some results on the strong convergence of weighted sums, including results on almost surely (a.s.) and complete convergence. To this end, we verified some Borel–Cantelli lemmas of the widely dependent random variables. The above-mentioned random variables contain common negatively dependent random variables, some positively dependent random variables, and some others; therefore, the obtained results extend and improve some existing results.     

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.     

Complete convergence for arrays of rowwise negatively superadditive-dependent random variables and its applications

In this paper, the Rosenthal-type maximal inequalities and Kolmogorov-type exponential inequality for negatively superadditive-dependent (NSD) random variables are presented. By using these inequalities, we study the complete convergence for arrays of rowwise NSD random variables. As applications, the Baum–Katz-type result for arrays of rowwise NSD random variables and the complete consistency for the estimator of nonparametric regression model based on NSD errors are obtained. Our results extend and improve the corresponding ones of Chen et al. [On complete convergence for arrays of rowwise negatively associated random variables. Theory Probab Appl. 2007;52(2):393–397] for arrays of rowwise negatively associated random variables to the case of arrays of rowwise NSD random variables.     

Lp convergence and complete convergence for weighted sums of AANA random variables

Let {X_n, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {d_ni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the L^p convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.     

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.     

WEIGHTED SUMS OF NEGATIVELY ASSOCIATED RANDOM VARIABLES

In this paper, we establish strong laws for weighted sums of negatively associated (NA) random variables which have a higher‐order moment condition. Some results of Bai Z.D. & Cheng P.E. (2000) [Marcinkiewicz strong laws for linear statistics. Statist. and Probab. Lett. 43, 105–112,] and Sung S.K. (2001) [Strong laws for weighted sums of i.i.d. random variables, Statist. and Probab. Lett. 52, 413–419] are sharpened and extended from the independent identically distributed case to the NA setting. Also, one of the results of Li D.L. et al. (1995) [Complete convergence and almost sure convergence of weighted sums of random variables. J. Theoret. Probab. 8, 49–76,] is complemented and extended.