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In this work, the complete moment convergence and Lp convergence for asymptotically almost negatively associated (AANA, in short) random variables are investigated. As an application, the complete convergence theorem for weighted sums of AANA random variables is obtained. These theorems obtained extend and improve some earlier results.

In this paper, the complete convergence is established for the weighted sums of negatively superadditive-dependent random variables. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for the random weighted average is also achieved, and a simulation study is done for the asymptotic behaviour of random weighting estimator.

In this paper, some results on the complete moment convergence for weighted sums of weakly dependent (or ρ ∗ -mixing) random variables are established. The results obtained in this paper improve and extend the corresponding one of Sung (2010). As an application of the main results, we present a result on complete consistency for the weighted estimator in a nonparametric regression model based on ρ ∗ -mixing errors.

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).

In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

In this note sufficient conditions for relative weak compactness of sums centered by constants of pair-wise independent random variables and for sums of squares of random variables centered by their medians are given. These conditions become necessary and sufficient if random variables are independent. The conditions are inspired by classical conditions for weak convergence of sums of uniformly small random variables.

In this paper, we study complete convergence for arrays of rowwise widely orthant dependent (WOD) random variables. Some general results on complete convergence for arrays of rowwise WOD random var...

In this paper, we provide some probability and moment inequalities (especially the Marcinkiewicz-Zygmund type inequality) for extended negatively dependent (END, in short) random variables. By using the Marcinkiewicz-Zygmund type inequality and the truncation method, we investigate the complete convergence for sums and weighted sums of arrays of rowwise END random variables. In addition, the complete moment convergence for END random variables is obtained. Our results generalize and improve the corresponding ones of Wang et al. [18] and Baek and Park [2].

In this paper, the complete moment convergence and the integrability of the supremum for weighted sums of negatively orthant dependent (NOD, in short) random variables are presented. As applications, the complete convergence and the Marcinkiewicz-Zygmund type strong law of large numbers for NODrandom variables are obtained. The results established in the paper generalize some corresponding ones for independent random variables and negatively associated random variables.

In this paper, an exponential inequality for the maximal partial sums of negatively superadditive-dependent (NSD, in short) random variables is established. By using the exponential inequality, we present some general results on the complete convergence for arrays of rowwise NSD random variables, which improve or generalize the corresponding ones of Wang et al. [28] and Chen et al. [2]. In addition, some sufficient conditions to prove the complete convergence are provided. As an application of the complete convergence that we established, we further investigate the complete consistency and convergence rate of the estimator in a nonparametric regression model based on NSD errors.

In this paper, some complete convergence, complete moment convergence, and mean convergence results for arrays of rowwise asymptotically negatively associated (ANA) random variables are obtained. These theorems not only generalize some well-known ones to ANA cases, but they also improve them.

In this paper, we investigate the complete moment convergence and Lr 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.

In this article, the complete convergence for weighted sums of extended negatively dependent (END, for short) random variables is investigated. Some sufficient conditions for the complete convergence are provided. In addition, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of END random variables is obtained. The results obtained in the article generalise and improve the corresponding one of Wang et al. [(2014b), ‘On Complete Convergence for an Extended Negatively Dependent Sequence’, Communications in Statistics-Theory and Methods, 43, 2923–2937]. As an application, the complete consistency for the estimator of nonparametric regression model is established.

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 th...

In this work, we establish some exponential inequalities for widely orthant-dependent random variables. We also obtain the convergence rate O(n− 1/2ln 1/2n) for the strong law of large numbers for widely orthant-dependent random variables.

In this paper, a useful inequality for central moment of uncertain random variables is proved. Based on this inequality, a convergence theorem for sum of uncertain random variables is derived. A Borel---Cantelli lemma for chance measure is obtained based on the continuity assumption of uncertain measure. Finally, several convergence theorems for uncertain random sequences are established.

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) and Gut and Stadtmuller (2011).

Some maximal inequalities for quadratic forms of independent and linearly negative quadrant dependent fuzzy random variables are established. Strong convergence of such quadratic forms are proved based on the martingale theory. A weak law of large numbers for linearly negative quadrant dependent fuzzy random variables is stated and proved.

The purpose of this paper is to establish the complete moment convergence for nonstationary negatively associated random variables satisfying the weak mean domination condition. The result is an improvement of complete convergence in Marcinkiewicz-Zygmund-type SLLN for negatively associated random variables in Kuczmaszewska (Acta Math. Hung. 128:116-130, 2010).

In this paper, we study infinite products of independent random operators. Conditions for the existence a.s. and the convergence in mean of order p of such infinite products are established.

New conditions are found for the absolute convergence of series of random variables almost surely. The results contain no independence assumptions. A generalization in analytical terms is obtained. Bibliography: 5 titles.

In this article, we investigate complete convergence and complete moment convergence for weighted sums of arrays of rowwise asymptotically negatively associated (ANA) random variables. The results obtained not only generalize the corresponding ones of Sung (Stat. Pap. 52:447-454, 2011), Zhou et al. (J. Inequal. Appl. 2011:157816, 2011), and Sung (Stat. Pap. 54:773-781, 2013) to the case of ANA random variables, but also improve them.

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.

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.

The concept of negative dependence for fuzzy random variables is introduced. The basic properties of such random variables are investigated. Some results on weak and strong convergence for sums and weighted sums of pairwise negatively dependent fuzzy random variables are derived. As a direct extension of classical methods, some limit theorems are established based on the concept of variance and covariance.

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).

In this paper, the complete convergence and complete moment convergence for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables are investigated. Some sufficient conditions to prove the complete convergence and the complete moment convergence are presented. The results obtained in the paper generalize and improve some corresponding ones for independent random variables and negatively associated random variables.

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.

We revisit the proofs of convergence for a first order primal---dual algorithm for convex optimization which we have studied a few years ago. In particular, we prove rates of convergence for a more general version, with simpler proofs and more complete results. The new results can deal with explicit terms and nonlinear proximity operators in spaces with quite general norms.

We remark the incorrectness of some recent results concerning Λ2-strong convergence. We give a new appropriate definition for the Λ2-strong convergence by generalizing the original Λ-strong convergence concept given by F. Moricz.

We study the rate of convergence in the strong law of large numbers expressed in terms of complete convergence of Baum–Katz type for sequences of random variables satisfying Petrov’s condition.

The authors study the complete moment convergence of weighted sums for arrays of rowwise negatively dependent random variables. The obtained results improve the corresponding results of Baek and Park (2010). Convergence of weighted sums for arrays of negatively dependent random variables and its applications. As an application, the authors obtain the complete moment convergence of linear processes based on pairwise negatively dependent random variables. In addition, the authors point out a gap of the proof in Baek and Park (2010) and raise an open problem.

Under some simple conditions, the convergence of a generalized chaotic system about its all variables is derived by only considering the convergence of its partial variables. Furthermore, based on some inequality techniques and employing the Lyapunov method, some novel sufficient criteria are derived to ensure the state variables of the discussed mixed delay system to converge, globally exponentially to a ball in the state space with a pre-specified convergence rate. Meanwhile, the ultimate bounds of the generalized chaotic system about its all variables are induced by the ultimate bounds of the system about its partial variables. Moreover, the maximum convergence rates about partial variables are also given. The methods are simple and valid for the convergence analysis of systems with time-varying and finite distributed delays. Here, the existence and uniqueness of the equilibrium point needs not to be considered. These simple conditions here are easy to be verified in engineering applications. Finally, some illustrated examples are given to show the effectiveness and usefulness of the results.

In this paper, the equivalent conditions of complete moment convergence of the maximum partial weighted sums for negatively superadditive dependent (NSD) random variables are established without the assumption of identical distribution. As applications, the complete moment convergence, the complete convergence and strong law of large numbers for NSD random variables are obtained. The results obtained in the paper generalize or improve the corresponding ones for weighted sums of independent random variables and negatively associated random variables.

This paper deals with the problem of almost sure convergence rate for weighted sums of negatively associated random variables. A new convergence rate is obtained base on an exponential inequality, the result obtained extends and has a fast convergence rate compare with the existing result. As an application, we study the Priestley-Chao estimate of nonparametric regression estimate and the convergence rate is derived.

In this paper, some limit theorems for arrays of rowwise pairwise negatively quadratic dependent random variables are studied. The open problem of [H. S. Sung, S. Lisawadi, and A. Volodin, J. Korean Math. Soc., 45 (2008), pp. 289--300] is solved. In addition, $p$-mean convergence and complete convergence under some appropriate conditions are studied.

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet–Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener–Poisson space. In the former space, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. In the latter space we give sufficient conditions for a sequence of multiple (Wiener–Poisson) integrals to converge to a Normal random variable.

In this paper, we obtain strong convergence property for Jamison weighted sums of negatively superadditive dependent (NSD, in short) random variables, which extends the famous Jamison theorem. In addition, some sufficient conditions for complete convergence for weighed sums of NSD random variables are presented. These results generalize the corresponding results for independent identically distributed random variables to the case of NSD random variables without assumption of identical distribution.

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), Sung (2012), and Wu (2010) can be obtained.

In this paper, we study the complete convergence and complete moment convergence for weighted sums of extended negatively dependent (END) random variables under sub-linear expectations space with the condition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\mathbb{V}}[|X|^{p}l(|X|^{1/\alpha})]0$\end{document}l(x)>0 is a slow varying and monotone nondecreasing function). As an application, the Baum-Katz type result for weighted sums of extended negatively dependent random variables is established under sub-linear expectations space. The results obtained in the article are the extensions of the complete convergence and complete moment convergence under classical linear expectation space.

We investigate the complete convergence of partial sums of randomly weighted extended negatively dependent (END) random variables. Some results of complete moment convergence, complete convergence and the strong law of large numbers for this dependent structure are obtained. As an application, we study the convergence of the state observers of linear-time-invariant systems. Our results extend the corresponding earlier ones.

Adding and removing redundant clauses is at the core of state-of-the-art SAT solving. Crucial is the ability to add short clauses whose redundancy can be determined in polynomial time. We present a characterization of the strongest notion of clause redundancy (i.e., addition of the clause preserves satisfiability) in terms of an implication relationship. By using a polynomial-time decidable implication relation based on unit propagation, we thus obtain an efficiently checkable redundancy notion. A proof system based on this notion is surprisingly strong, even without the introduction of new variables—the key component of short proofs presented in the proof complexity literature. We demonstrate this strength on the famous pigeon hole formulas by providing short clausal proofs without new variables.

In this paper, the complete qth moment convergence of weighted sums for arrays of rowwise negatively associated (NA) random variables is investigated. By using moment inequality and truncation methods, some general results on complete qth moment convergence of weighted sums for arrays of rowwise NA random variables are obtained. As their applications, we not only generalize and extend the corresponding results of Baek et al. [On the complete convergence of weighted sums for arrays of negatively associated variables, J. Korean Stat. Soc. 37 (2008), pp. 73–80], Liang [Complete convergence for weighted sums of negatively associated random variables, Stat. Probab. Lett. 48 (2000), pp. 317–325 and Liang et al. [Complete moment convergence for sums of negatively associated random variables, Acta Math. Sin. English Ser. 26 (2010), pp. 419–432], but also greatly simplify their proofs.

It is known that the dependence structure of pairwise negative quadrant dependent (NQD) random variables is weaker than those of negatively associated random variables and negatively orthant dependent random variables. In this article, we investigate the moving average process which is based on the pairwise NQD random variables. The complete moment convergence and the integrability of the supremum are presented for this moving average process. The results imply complete convergence and the Marcinkiewicz–Zygmund-type strong law of large numbers for pairwise NQD sequences.

In this article, we study the complete convergence for non-stationary ϕ-mixing random variables, especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for ϕ-mixing random variables. Our result generalizes the corresponding one of Shao (1988) and improves the corresponding one of Peligrad (1985a) and Wang (1987).

We present new results on the Hamburger moment problem for probability distributions and apply them to characterize the moment determinacy of powers and products of i.i.d. random variables with values in the whole real line. Detailed proofs of all results are given followed by comments and examples. We also provide new and more transparent proofs of a few known results. E.g., we give a new and short proof that the product of three or more i.i.d. normal random variables is moment-indeterminate. The illustrations involve specific distributions such as the double generalized gamma (DGG), normal, Laplace and logistic. We show that sometimes, but not always, the power and the product of i.i.d. random variables (of the same odd ‘order’) share the same moment determinacy property. This is true for the DGG and the logistic distributions. The paper also treats two unconventional types of problems: products of independent random variables of different types and a random power of a given random variable. In particular, we show that the product of Laplace and logistic random variables, the product of logistic and exponential random variables, the product of normal and χ2 random variables, and the random power ZN, where Z ~ N and N is a Poisson random variable, are all moment-indeterminate.

In the paper, we generalize the von Bahr–Esseen moment inequality from independent random variables to pairwise independent random variables. As the applications, the moment convergence, the complete convergence and the strong law of large numbers are established for pairwise independent random variables.