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]).     

Almost everywhere convergence for sequences of pairwise NQD random variables

In this article, we are going to study the almost everywhere convergence for sequences of pairwise negatively quadrant dependent random variables by using truncation technique and Kolmogorov-type generalized three-series theorem. Our results generalize and improve the corresponding results of Wu (2002 Wu, Q. Y. (2002). Convergence properties of pairwise NQD random sequence. Acta. Math. Sin. 45:617–624 (in Chinese). [Google Scholar]) and Li and Yang (2008 Li, R., Yang, W. G. (2008). Strong convergence of pairwise NQD random sequences. J. Math. Anal. Appl. 334:741–747.[Crossref], [Web of Science ®] , [Google Scholar]). We also give some examples showing that our extensions are not trivial.     

Complete convergence of the non-identically distributed pairwise NQD random sequences

ABSTRACT

In this article, we study complete convergence of the nonidentically distributed pairwise negatively quadrant dependent (NQD) random sequences by the moment inequality and terminating random variables,which extend and improve the previous relevant results.     

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.     

Complete convergence and the strong laws of large numbers for pairwise NQD random variables

The authors study the strong convergence for sequences of pairwise negatively quadrant dependent (NQD) random variables under some wide conditions, and present some new theorems on the complete convergence and the strong laws of large numbers. The obtained results extend and improve some theorems in existing literature.     

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.     

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.     

Strong Limit Theorems for Pairwise NQD Random Variables

A number of strong laws of large numbers for sequences of pairwise negative quadrant dependent (NQD) random variables have been established by using the generalized three series theorem. In this article, we obtain a strong law of large numbers by using the truncation technique and method of subsequences instead of the generalized three series theorem. Our result generalizes and improves on the corresponding one in Li and Yang (2008 Li , R. , Yang , W. ( 2008 ). Strong convergence of pairwise NQD random sequences . J. Math. Anal. Appl. 344 : 741 – 747 .[Crossref], [Web of Science ®] , [Google Scholar]). We also obtain a complete convergence result for an array of rowwise pairwise NQD random variables.     

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.     

Strong laws for associated random variables

We study the almost sure convergence and rates of weighted sums of associated random variables under the classical assumption of existence of Laplace transforms. This assumption implies the existence of every moment, so we address the same problem assuming a suitable decrease rate on tail joint probabilities which only implies the existence of finitely many moments, proving the analogous characterizations of convergence and rates. Still relaxing further the assumptions on moment existence, we also prove a Marcinkiewicz–Zygmund for associated variables without means, complementing existing results for this dependence structure.     

Complete convergence for weighted sums of arrays of APND random variables

In this note, we introduce a new class of dependent random variables (henceforth rvs), together with some its basic properties. This class includes independent rvs and pairwise negatively dependent rvs. Some extensions of Ranjbar et al. (2008) are discussed. The complete convergence for the new class of rvs is also proved, and some results of Beak and Park (2010 Beak, J.-II., and S. T. Park. 2010. Convergence of weighted sums for arrays of negatively dependent random variables and its applications. J. Stat. Plann. Inference 140:2461–2469.[Crossref], [Web of Science ®] , [Google Scholar]) are extended to this class conveniently.     

Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model

In this paper, we obtain complete convergence results for Stout type weighted sums of i.i.d. random variables. A strong law for weighted sums of i.i.d. random variables is also obtained. As the applications of the strong law, the strong consistency and rate of the nonparametric regression estimations and the rates of the strong consistency of LS estimators for the unknown parameters of the simple linear errors in variables (EV) model are given.     

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 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 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 arrays of rowwise negatively superadditive dependent random variables

In this paper, some complete convergence and complete moment convergence results for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables are studied. The obtained theorems not only extend the result of Gan and Chen (2007 Gan, S. X., and P. Y. Chen. 2007. On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables. Acta Mathematica Scientia. Series B 27 (2):283–90.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of NSD random variables, but also improve them.     

Complete and complete moment convergence for i.i.d. random variables under exponential moment conditions

In this paper, we establish a complete convergence result and a complete moment convergence result for i.i.d. random variables under moment condition which is slightly weaker than the existence of the moment generating function. The main results extend and improve the related known results of Lanzinger (1998 Lanzinger, H. (1998). A Baum-Katz theorem for random variables under exponential moment conditions. Stat. Probab. Lett. 39(2):89–95.[Crossref], [Web of Science ®] , [Google Scholar]) and Gut and Stadtmüller (2011 Gut, A., Stadtmüller, U. (2011). An intermediate Baum-Katz theorem. Stat. Probab. Lett. 81(10):1486–1492.[Crossref], [Web of Science ®] , [Google Scholar]).     

Complete convergence for weighted sums of negatively dependent random variables under the sub-linear expectations

In this article, we study complete convergence theorems for weighted sums of negatively dependent random variables under the sub-linear expectations. Our results extend the corresponding results of Sung (2012 Sung, S. H. 2012. A note on the Complete convergence for weighted sums of negatively dependent random variables. Journal of Inequalities and Applications 2012:158, 10 pages. [Google Scholar]) relative to the classical probability.