  This note develops Rio's proof [C. R. Math. Acad. Sci. Paris, 1995] of the
rate of convergence in the Marcinkiewicz--Zygmund strong law of large numbers
to the case of sums of dependent random variables with regularly varying
normalizing constants. It allows us to obtain a complete convergence result for
dependent sequences under uniformly bounded moment conditions. This result is
new even when the underlying random variables are independent. The main
theorems are applied to three different dependence structures: (i) $m$-pairwise
negatively dependent random variables, (ii) $m$-extended negatively dependent
random variables, and (iii) $\varphi$-mixing sequences. To our best knowledge,
the results for cases (i) and (ii) are the first results in the literature on
complete convergence for sequences of $m$-pairwise negatively dependent random
variables and $m$-extended negatively dependent random variables under the
optimal moment conditions even when $m=1$. While the results for cases (i) and
(iii) unify and improve many existing ones, the result for case (ii)
complements the main result of Chen et al. [J. Appl. Probab., 2010].
Affirmative answers to open questions raised by Chen et al. [J. Math. Anal.
Appl., 2014] and Wu and Rosalsky [Glas. Mat. Ser. III, 2015] are also given. An
example illustrating the sharpness of the main result is presented.
