Exponential Inequalities For NOD Sequence And Its Application

Let{Zn,n≥1} be a sequence of independent nonnegative random variables with finite second moments. Denote We will show that under suitable conditions the following equivalence relation holds, namely, where a≥0 andα> 0 are arbitrary real numbers. Here and below, cn～dn means Garcia and Palacios (2001) pointed out that people needed to calculate the inverse moment such as E{(1+Xn))-a), and discussed the sufficient condition of the equality following If the equality above is right, then for all n sufficiently large.In generally, the left type of the inverse moment is difficult to calculate, but easy to get the right type of value. Under certain asymptotic-normality condition, relation (1.2) (a= 1) is established in Theorem 2.1 of Garcia and Palacios (2001). But, unfortunately, that theorem was not true under the suggested assumptions, as pointed out by Kaluszka and Okolewski (2004). The latter authors established (1.2) by modifying the assumptions, as follows:(i)α<3 (α<4, in the i.i.d. case);(ii) EXn→∞, EZn3<∞; (iii) (Lc condition)Hu, et al. (2007) considered weaker conditions:EZn2+δ<∞, where Zn satisfies L2+δcondition and 0<δ≤1. Recently, Wu, et al. (2009) applied Bernstein's inequality and the truncated method to improve the conclusion in weaker condition on moment and obtained the following result: Theorem 1.1. Suppose{Zn,n≥1} is a sequence of independent, nonnegative and non-degenerated random variables such that EZn2<∞for all n and EXn→∞, where Xn is defined by (1.1). Furthermore, if we assume that there exists a finite positive constant C, not depending on n such that for someη> 0, then relation (1.2) holds for all real numbers a> 0 andα> 0.The main results above are studied under the case of independent sequence random variables. Can they be extended to the dependent sequence? Therefore, it has a certain theoretical and practical value of research this problem. Inspired by Hoeffding (1963), Christofides and Hadjikyriakou (2009) and Wu, etc. (2009), we not only obtain some the exponential inequalities containing Bernstein's inequality for NOD sequence and complete convergence of sums, but also give the asymptotic approximation of inverse moments for non-negative NOD sequence, which generalizes and improves the results of Theorem 3 in Kaluszka and Okolewski (2004), Theorem 2.1 and Theorem 2.3 in Hu, et al. (2007) and Theorem 1 in Wu, et al. (2009).