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

For a sequence of i.i.d. random variables { X , X n , n ≥ 1 } and a sequence of positive real numbers { a n , n ≥ 1 } with 0 < a n / n 1 / p ↑ for some 0 < p < 2 , the Baum-Katz complete convergence theorem is extended to the { X , X n , n ≥ 1 } with the general moment condition ∑ n = 1 ∞ n r − 1 P { | X | > a n } < ∞ , where r ≥ 1 . The relationship between the complete convergence and the strong law of large numbers is established.