The concept complete convergence was introduced in 1947 by Hsu and Robbins who proved that the sequence of arithmetic means of i.i.d. random variables converges completely to the expected value of the variables provided their variance is finite. The complete convergence of dependent random variables has been investigated by several authors, for example, Amini M. and Bozorgnia A. (2003), Li, Y.X. and Zhang, L. (2004), Chen, P. et al. (2007), Amini, M. et al. (2007), Wu, Q.Y. (2010), Ko, M.H. (2011), Amini, M. et al. (2012), Wang, X. et al. (2012), Yang, W. et al. (2012), Sung, S.H. (2012), Shen, A.T. et al.(2013) Wang, X. et al. (2014), Amini, M. et al. (2015), Wang, X. et al. (2015), Amini, M. et al. (2016), Deng, X. et al. (2016), and Amini, M. et al. (2017). In general, the main tools to prove the complete convergence of some random variables are based on Borel-Cantelli lemma and the moment inequality or the exponential inequality. However, for some dependent sequences (such as weakly negative dependent (WND) and negative superadditivedependent random (NSD) sequence), whether these inequalities hold was not known. In this talk, we review complete convergence as historically from i.i.d. sequences to dependent sequences. In particular, complete convergence for weighted sums of weakly negative dependent are provided and applied to empirical distribution, sample p-th quantile and random weighting estimate. Also, the complete convergence is established for weighted sums of negatively superadditive-dependent random variables. Moreover, under the condition of integrability and appropriate conditions on the array of weights, the conditional mean convergence and conditional almost sure convergence theorems for weighted sums of an array of random variables are obtained when the random variables are special kind of dependence.

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