Aiting Shen

Probability inequalities for END sequence and their applications

Some probability inequalities for extended negatively dependent (END) sequence are provided. Using the probability inequalities, we present some moment inequalities, especially the Rosenthal-type inequality for END sequence. At last, we study the asymptotic approximation of inverse moment for nonnegative END sequence with finite first moments, which generalizes and improves the corresponding results of Wu et al. [Stat. Probab. Lett. 79, 1366-1371 (2009)], Wang et al. [Stat. Probab. Lett. 80, 452-461 (2010)], and Sung [J. Inequal. Appl. 2010, Article ID 823767, 13pp. (2010). doi:10.1155/2010/823767].MSC(2000): 60E15; 62G20.

Some strong limit theorems for arrays of rowwise negatively orthant-dependent random variables

In this article, the strong limit theorems for arrays of rowwise negatively orthant-dependent random variables are studied. Some sufficient conditions for strong law of large numbers for an array of rowwise negatively orthant-dependent random variables without assumptions of identical distribution and stochastic domination are presented. As an application, the Chung-type strong law of large numbers for arrays of rowwise negatively orthant-dependent random variables is obtained.MR(2000) Subject Classification: 60F15

Complete moment convergence for arrays of rowwise NSD random variables

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.

On the Rate of Complete Convergence for Weighted Sums of Arrays of Rowwise ϕ-Mixing Random Variables

Let be an array of rowwise ϕ-mixing random variables. A rate of complete convergence for weighted sums of arrays of rowwise ϕ-mixing random variables is obtained without assumption of identical distribution. The techniques used in the paper are the Rosenthal type inequality and the truncated method. As an application, the Baum and Katz type result for arrays of rowwise ϕ-mixing random variables is obtained.

On Complete Convergence for Nonstationary ϕ-Mixing Random Variables

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).

Conditional convergence for randomly weighted sums of random variables based on conditional residual h-integrability

Let {Xnk,un≤k≤vn,n≥1} and {Ank,un≤k≤vn,n≥1} be two arrays of random variables defined on the same probability space (Ω,A,P) and Bn be sub-σ-algebras of A. Let r>0 be a constant. In this paper, we introduce some concepts of conditional residual h-integrability such as conditionally residually h-integrable relative to Bn concerning the array {Ank} with exponent r and conditionally strongly residually h-integrable relative to Bn concerning the array {Ank} with exponent r. These concepts are more general than some known setting of randomly weighted sums of random variables. Based on the conditions of conditional residual h-integrability with exponent r and conditional strongly residual h-integrability with exponent r, we obtain the conditional mean convergence and conditional almost sure convergence for randomly weighted sums.MSC:60F15, 60F25.

Some probability inequalities for a class of random variables and their applications

Some probability inequalities for a class of random variables are presented. As applications, we study the complete convergence for it. Our main results generalize the corresponding ones for negatively associated random variables and negatively orthant dependent random variables.MSC:60E15, 60F15.

Complete Convergence of the Maximum Partial Sums for Arrays of Rowwise of AANA Random Variables

The limiting behavior of the maximum partial sums is investigated, and some new results are obtained, where is an array of rowwise AANA random variables and is a sequence of positive real numbers. As an application, the Chung-type strong law of large numbers for arrays of rowwise AANA random variables is obtained. The results extend and improve the corresponding ones of Hu and Taylor (1997) for arrays of rowwise independent random variables.

Complete convergence for weighted sums of arrays of rowwise ˜ ρ-mixing random variables

Let {Xni,i≥1,n≥1} be an array of rowwise ρ˜-mixing random variables. Some sufficient conditions for complete convergence for weighted sums of arrays of rowwise ρ˜-mixing random variables are presented without assumptions of identical distribution. As applications, the Baum and Katz type result and the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of ρ˜-mixing random variables are obtained.MSC:60F15.

Convergence properties for weighted sums of NSD random variables

Abstract Let {Xn, n ⩾ 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {bni, 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 ∑ni = 1bniXi 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.