Starting with independent, identically distributed random variables X
2... and their partial sums (S
), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑
>b(n)) and aim for necessary and sufficient conditions on X
1 in order to obtain the existence of certain moments for N, as well as for generalized counting variables with weights, and multi-index random variables. The existence of the first moment of N when b(n)=εn, i.e. ∑
|>εn)<∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.