A strong law of large number for negatively dependent and non identical distributed random variables in the framework of sublinear expectation

Abstract

In this article, in the framework of sublinear expectation initiated by Peng, we derive a strong law of large numbers (SLLN) for negatively dependent and non identical distributed random variables. This result includes and extends some existing results. Furthermore, we give two examples of our result for applications.     

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 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.     

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.     

General strong convergence for negatively associated random variables

For negatively associated (NA) random variables, we obtain two general strong laws of large numbers (SLLN) in which the coefficient of sum and the weight are both general functions. As corollaries, we obtain Marcinkiewicz-type SLLN, the logarithmic SLLN and Marcinkiewicz SLLN for NA random variables.     

General laws of large numbers under sublinear expectations

ABSTRACT

In this paper, under some weaker conditions, we give three laws of large numbers (LLNs) under sublinear expectations (capacities), which extend the LLN under sublinear expectations in Peng (2008b Peng, S. (2008b). A new central limit theorem under sublinear expectations. arXiv:0803.2656v1 [math.PR], 18 Mar 2008. [Google Scholar]) and the strong LLN for capacities in Chen (2010 Chen, Z. (2010). Strong laws of large numbers for capacities. arXiv:1006.0749v1 [math.PR], 3 Jun 2010. [Google Scholar]). It turns out that these theorems are natural extensions of the classical strong (weak) LLNs to the case where probability measures are no longer additive.     

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 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.     

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 convergence for weighted sums of negatively dependent random variables

In this paper, we obtain a complete convergence result for weighted sums of negatively dependent random variables under mild conditions of weights. This result generalizes and improves the result of Zarei and Jabbari (Stat Papers doi:, 2009). Our result also extends the result of Taylor et al. (Stoch Anal Appl 20:643–656, 2002) on unweighted average to a weighted average.     

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.     

On limiting behavior for arrays of rowwise negatively orthant dependent random variables

In this paper, the authors study limiting behavior for arrays of rowwise negatively orthant dependent random variables and obtain some new results which extend and improve the corresponding theorems by Hu, Móricz, and Taylor (1989), Taylor, Patterson, and Bozorgnia (2002) and Wu and Zhu (2010).     

The Discounted Large Deviation Principle for Autoregressive Processes

In this article, the discounted large deviation principle for autoregressive processes is obtained under the Bernstein condition.     

Exponential inequality for negatively associated random variables

Under mild conditions, a Bernstein-Hoeffding-type inequality is established for covariance invariant negatively associated random variables. The proof uses a truncation technique together with a block decomposition of the sums to allow an approximation to independence.