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Let {Xn} n≥1 be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centered sequence with a suitable level of negative dependence. We prove Baum-Katz type theorems by only assuming that the variables Xn satisfy a uniform moment bound condition. We also prove that this condition is best possible even for sequences of centered, independent random variables. This leads to Marcinkiewicz-Zygmund type strong laws of large numbers with estimate for the rate of convergence.By the Borel-Cantelli Lemma this implies that X n → 0 almost surely, but the converse is not necessarily true. If {X n } n≥1 is a centered i.i.d. sequence of random variables then S n /n → 0 almost surely by the strong law of large numbers. Under what conditions does S n /n converge completely to 0? Hsu and Robbins [13] showed that E(X 2 1 ) < ∞ is sufficient, and Erdős [10,11] proved that it is necessary.2010 Mathematics Subject Classification. Primary: 60F15, 60G42, 60G50; Secondary: 60F10. Key words and phrases. complete convergence, Marcinkiewicz-Zygmund strong law of large numbers, rate of convergence, independent random variables, martingale difference sequences.

Pairwise independent random variables with multidimensional indices are studied. The Kolmogorov and the Marcinkiewicz strong laws of large numbers and Spitzer's theorem are proved in the case of exponent r ≤ 1.

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