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.     

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]).     

Complete moment convergence of widely orthant dependent random variables

It is known that the dependence structure of widely orthant dependent (WOD) random variables is weaker than those of negatively associated (NA) random variables, negatively superadditive dependent (NSD) random variables, negatively orthant dependent (NOD) random variables, and extended negatively dependent (END) random variables. In this article, the results of complete moment convergence and complete convergence are presented for WOD sequence under the same moment conditions as independent sequence in classical result (Chow 1988 Chow, Y. (1988). On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16(3):177–201. [Google Scholar]).     

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 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.     

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.     

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 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.     

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.     

On asymptotic approximation of inverse moments for a class of nonnegative random variables

Sung [On inverse moments for a class of nonnegative random variables. J Inequal Appl. 2010;2010:1–13. Article ID 823767, doi:10.1155/2010/823767] obtained the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite second moments and satisfying a Rosenthal-type inequality. In the paper, we further study the asymptotic approximation of inverse moments for a class of nonnegative random variables with finite first moments, which generalizes and improves the corresponding ones of Wu et al. [Asymptotic approximation of inverse moments of nonnegative random variables. Statist Probab Lett. 2009;79:1366–1371], Wang et al. [Exponential inequalities and inverse moment for NOD sequence. Statist Probab Lett. 2010;80:452–461; On complete convergence for weighted sums of ? mixing random variables. J Inequal Appl. 2010;2010:1–13, Article ID 372390, doi:10.1155/2010/372390], Sung (2010) and Hu et al. [A note on the inverse moment for the nonnegative random variables. Commun Statist Theory Methods. 2012. Article ID 673677, doi:10.1080/03610926.2012.673677].     

On convergence rate in the SLLN for maximums of moving-average sums of ALNQD random fields

In this article, we define a notion of asymptotically linear negatively quadrant dependence and establish the rate of complete convergence for maximums of moving-average sums of asymptotically linear negatively quadrant dependent random fields.     

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.     

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.     

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.     

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.     

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].     

A limit theorem on moment convergence of centered spectral statistics of random matrices

ABSTRACT

The eigenvalues of a random matrix are a sequence of specific dependent random variables, the limiting properties of which are one of interesting topics in probability theory. The aim of the article is to extend some probability limiting properties of i.i.d. random variables in the context of the complete moment convergence to the centered spectral statistics of random matrices. Some precise asymptotic results related to the complete convergence of p-order conditional moment of Wigner matrices and sample covariance matrices are obtained. The proofs mainly depend on the central limit theorem and large deviation inequalities of spectral statistics.