In this paper by using the equivalence of the a.s. convergence and the convergence in probability for the partial sums of independent random variables, we improved and strengthened a series of M. Katz
and L. E. Baum
’s results [Trans. Am. Math. Soc. 120, 108-123 (1965; Zbl 0142.148
)], which give some necessary and sufficient conditions about the complete convergence for the partial sums of iid random variables. On the above basis, we got ideal conditions and results about the complete convergence for the random selected partial sums of iid random variables.