Strong laws of large numbers for arrays of rowwise independent random elements - Robert Lee Taylor; Tien-Chung Hu (International Journal of Mathematics and Mathematical Sciences)
Let Xnk be an array of rowwise independent random elements in a separable Banach space of type p+δ with EXnk=0 for all k, n. The complete convergence (and hence almost sure convergence) of n−1/p∑k=1nXnk to 0, 1≤p<2, is obtained when Xnk are uniformly bounded by a random variable X with E|X|2p<∞. When the array Xnk consists of i.i.d, random elements, then it is shown that n−1/p∑k=1nXnk converges completely to 0 if and only if E‖X11‖2p<∞.