On the Almost Sure Convergence Rates for Pairwise Negative Quadrant Dependent Random Variables

For sequences of pairwise negative quadrant dependent random variables, we obtain the almost sure convergence rate n(-1/2) (log n)(3/2) (log log n)(1/xi) with any 0 < xi < 2 of Sigma(n)(i=1) (X-i - EXi)/n -> 0 a.s. by a maximal moment inequality, which improves the relevant results in Wu (2002) and Wang et al. (2008). In addition, the faster convergence rate n(-1/2) (log n)(1/2) is also obtained by the exponential inequality established in this paper, which reaches the available one for independent random variables in terms of Berstein type inequality. Further, we give the corresponding precise asymptotic with respect to the rate n(-1/2)(log n)(1/2).