Baum–Katz type theorems with exact threshold

Let {Xn} n=1 be either a sequence of arbitrary random variables, or a martingale difference sequence, or a centred sequencewith a suitable level of negative dependence. Weprove 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 centred, independent random variables. This leads to Marcinkiewicz- Zygmund type strong laws of large numbers with estimate for the rate of convergence.