Strong convergence properties for partial sums of asymptotically negatively associated random vectors in Hilbert spaces

Abstract

In this paper, we investigate the almost sure convergence for partial sums of asymptotically negatively associated (ANA, for short) random vectors in Hilbert spaces. The Khintchine-Kolmogorov type convergence theorem, three series theorem and the Kolmogorov type strong law of large numbers for partial sums of ANA random vectors in Hilbert spaces are obtained. The results obtained in the paper generalize some corresponding ones for independent random vectors and negatively associated random vectors in Hilbert spaces.     

Probability inequalities for sums of NSD random variables and applications

Abstract

In this paper, we obtain the exponential-type inequalities for maximal partial sums of negatively superadditive dependent (NSD) random variables, which extends the corresponding results for independent and negatively associated (NA) random variables. Using these inequalities, we further investigate the weak convergence of the M-estimators in the generalized linear model with NSD errors, which generalize and improve the corresponding results of the independent random errors to that of NSD random errors.     

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.     

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.     

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.     

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.     

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.     

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 coordinatewise asymptotically negatively associated random vectors in Hilbert spaces

In this article, we study the complete convergence for sequences of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. We also investigate that some related results for coordinatewise negatively associated random vectors in Huan, Quang, and Thuan (2014 Huan, N. V., N. V. Quang, and N. T. Thuan. 2014. Baum–Katz type theorems for coordinatewise negatively associated random vectors in Hilbert spaces. Acta Mathematica Hungarica 144(1):132–419.[Crossref], [Web of Science ®] , [Google Scholar]) still hold under this concept.     

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 the rate of convergence in the strong law of large numbers for negatively orthant-dependent random variables

ABSTRACT

In this paper, we study the complete convergence and complete moment convergence for negatively orthant-dependent random variables. Especially, we obtain the Hsu–Robbins-type theorem for negatively orthant-dependent random variables. Our results generalize the corresponding ones for independent random variables.     

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.     

On Complete Convergence for Weighted Sums of NA Random Fields

In this paper, we prove the complete convergence for the weighted sums of negatively associated random variables with multidimensional indices. The main result generalizes Theorem 2.1 in Kuczmaszewska and Lagodowski (2011 Kuczmaszewska, A., Lagodowski, Z.A. (2011). Convergence rates in the SLLN for some classes of dependent random field. J. Math. Anal. Appl. 380:571–584.[Crossref], [Web of Science ®] , [Google Scholar]) to the case of weighted sums.     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

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.     

Complete and complete moment convergence with applications to the EV regression models

In this paper, the complete convergence and complete moment convergence for arrays of rowwise m-extended negatively dependent (m-END, for short) random variables are established. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for m-END random variables is also achieved. By using the results that we established, we further investigate the strong consistency of the least square estimator in the simple linear errors-in-variables models, and provide some simulations to verify the validity of our theoretical results.     

On almost sure convergence for weighted sums of pairwise negatively quadrant dependent random variables

Let {X _n , n ≥ 1} be a sequence of pairwise negatively quadrant dependent (NQD) random variables. In this study, we prove almost sure limit theorems for weighted sums of the random variables. From these results, we obtain a version of the Glivenko–Cantelli lemma for pairwise NQD random variables under some fragile conditions. Moreover, a simulation study is done to compare the convergence rates with those of Azarnoosh (Pak J Statist 19(1):15–23, 2003) and Li et al. (Bull Inst Math 1:281–305, 2006).     

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 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.     

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.