A strong law and a law of the single logarithm for arrays of rowwise independent random variables

Let 1≤p<2. In this paper, we show that there always exist arrays of rowwise independent random variables {Xnk,1≤k≤n,n≥1} with the same distribution as X, such that n−1/p∑k=1nXnk→0a.s . holds if and only if EX=0 and E|X|β<∞ for any β∈(p,2p). This says that the moment gap of the necessary and sufficient condition for the strong law of large numbers between the sequence (β=p) and the array (β=2p) is fulfilled. Analogous results are also obtained for the law of the single logarithm.