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For a sequence of i.i.d. random variables \(\{X, X_{n}, n\ge1\}\) and a sequence of positive real numbers \(\{a_{n}, n\ge1\}\) with \(0< a_{n}/n^{1/p}\uparrow\) for some \(0< p<2\), the Baum-Katz complete convergence theorem is extended to the \(\{X, X_{n}, n\ge1\}\) with the general moment condition \(\sum^{\infty}_{n=1}n^{r-1}P\{|X|>a_{n}\}<\infty\), where \(r\ge1\). The relationship between the complete convergence and the strong law of large numbers is established.

An extension of the Baum-Katz theorem to i.i.d. random variables with general moment conditions