On the exponential inequalities for negatively dependent random variables

Abstract

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.

Equivalent Conditions of Complete Moment Convergence and Complete Integral Convergence for Nod Sequences

“…A number of useful results for NOD random variables have been established by many authors. We refer to Volodin [19] for the Kolmogorov exponential inequality, Asadian et al [1] for Rosenthal's type inequality, Zarei and Jabbari [28], Wu [24], Wang et al [20], Sung [18], Yi et al [27] and Chen and Sung [4] for complete convergence, Wang et al [21] and Sung [17] for exponential inequalities, Wu and Jiang [25] for the strong consistency of M estimator in a linear model, Shen [12,14] for strong limit theorems of weighted sums, Shen [15] for the asymptotic approximation of inverse moments, Wang and Si [22] for the complete consistency of estimator of nonparametric regression model, Qiu et al [11] and Wu and Volodin [26] for the complete moment convergence, and so on.…”

On the Bahadur representation of sample quantiles for widely orthant dependent sequences

“…They pointed out that NA random variables are NOD random variables, but the converse statement cannot always be true. Various results and examples of NOD and NA random variables can be found in [1], [11], [13], [16], [19], [21], [24], etc.…”

On the Strong Convergence and Complete Convergence for Pairwise NQD Random Variables

“…The concept of pairwise NQD random variables was introduced by Lehmann [8], which includes pairwise independent random sequence and some negatively dependent sequences, such as negatively associated sequences (see [9][10][11][12][13]), negatively orthant dependent sequences (see [9,[14][15][16][17][18]), and linearly negative quadrant dependent sequences (see [19][20][21]). Hence, studying the probability limiting behavior of pairwise NQD random variables and its applications in probability theory and mathematical statistics are of great interest.…”