Strong laws for sequences in the vicinity of the LIL

The present paper is devoted to strong laws of large numbers under moment conditions near those of the law of the iterated logarithm (LIL) for i.i.d sequences. More precisely, we wish to investigate possible limit theorems under moment conditions which are stronger than p for any p<2, in which case we know that there is a.s. convergence to 0, and weaker than EX2<∞, in which case the LIL holds.