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The present paper is devoted to complete convergence and the strong law of large numbers under moment conditions near those of the law of the single logarithm (LSL) for independent and identically distributed arrays. More precisely, we investigate limit theorems under moment conditions which are stronger than 
$2p$ for any 
$p<2$, in which case we know that there is almost sure convergence to 0, and weaker than 
$E\,X^{4}/(\log ^{+}|X|)^{2}<\infty$, in which case the LSL holds.
