Necessary and Sufficient Conditions for Complete Convergence in the Law of Large Numbers

Relationships between the growth of a sequence $N_k$ and conditions on the tail of the distribution of a sequence $X_l$ of i.i.d. mean zero random variables are given that are necessary and sufficient for $$\sum^\infty_{k=1} P\big\{\big|\frac{1}{N_k} \sum^{N_k}_{l=1} X_l\big| > \varepsilon\big\} < \infty.$$ The results are significant for distributions satisfying $E(|X_l|) < \infty$ but $E(|X_l|^\beta) = \infty$ for some $\beta > 1$. Necessary and sufficient conditions for the finiteness of sums of the form $$\sum^\infty_{n=1} \gamma (n)P\big\{\big|\frac{1}{n} \sum^n_{l=1} X_l\big| > \varepsilon\big\}$$ are obtained as a corollary.