On the strong law of large numbers for sequences of blockwise independent and blockwise p-orthogonal random elements in rademacher type p Banach spaces

For a sequence of random elements {Vn, n ≥1} taking values in a real separable Rademacher type p (1 ≤p ≤2) Banach space and positive constants bn↑∞, conditions are provided for the strong law of large numbers ∑ni=1Vi/bn→0 almost surely. We treat the following cases: (i) {Vnn ≥1} is blockwise independent with EVn=0, n≥1, and (ii) {Vn, n≥1} is blockwise p-orthogonal. The conditions for case (i) are shown to provide an exact characterization of Rademacher type p Banach spaces. The current work extends results of Móricz [12], Móricz et al. [13], and Gaposhkin [8]. Special cases of the main results are presented as corollaries and illustrative examples or counterexamples are provided.