Abstract: In this paper, some unsolved problems on conditions of the complete oonvergence of stationary mixing sequenoes \left\X_n\right\ are discussed. In this way some necessary conditions and some sufficient conditions for \sum_n=1^\infty \frac\ln nn p\left(\max _k< n\left|S_n\right|>\varepsilon n^\frac1p\right)<\infty, \quad \forall \varepsilon>0and\sum_n=1^\infty \frac1n p\left(\sup _k>n\left|\fracS_kk^\frac1p\right|>\varepsilon\right)< \infty, \quad \forall \varepsilon>0 ere proved, where 0< p< 2, S_k=\sum_j=1^k X_j, and \left\X_n\right\ satisfy certain mixing conditions.