Exponential inequalities and inverse moment for NOD sequence

Probability inequalities for END sequence and their applications

“…Our results will extend and improve the results of Wu et al [8], Wang et al [9], and Sung [11]. Now, we state and prove the results of asymptotic approximation of inverse moments for nonnegative END random variables.…”

“…Wang et al [9] pointed out that the condition (iii) in Theorem A can be removed and extended the result for independent random variables to the case of NOD random variables. Shi et al [10] obtained (4.2) for B n = 1 and pointed out that the existence of finite second moments is not required.…”

Some exponential inequalities for acceptable random variables and complete convergence

“…In Section 2, we will present some exponential inequalities for a sequence of acceptable random variables, such as Bernstein-type inequality, Hoeffding-type inequality. The Bernstein-type inequality for acceptable random variables generalizes and improves the corresponding results of Yang [9] for NA random variables and Wang et al [10] for NOD random variables. In Section 3, we will study the complete convergence for acceptable random variables using the exponential inequalities established in Section 2.…”

The consistency for estimator of nonparametric regression model based on NOD errors

“…where c p depends only on p. [20]). Let {X n } n≥1 be a sequence of NOD random variables such that EX n = 0 and |X n | ≤ b for each n ≥ 1, where b is a positive constant.…”

“…They pointed out that NA random variables are NOD random variables, but neither NUOD nor NLOD implies NA. Various results and examples of NOD random variables can be found in Joag-Dev and Proschan [17], Bozorgnia et al [18], Asadian et al [19], Wang et al [20], Wu [21,22], Wang et al [23,24], Li et al [25] and Sung [26], etc. Obviously, by the definitions of NOD and pairwise NQD, NOD random variables are pairwise NQD random variables.…”