LAUSR

For a double array {Vm,n,m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space and an increasing double array {bm,n,m ≥ 1, n ≥ 1} of positive constants, the limit law $$max_{1\leq{k}\leq{m},1\leq{l}\leq{n}}\parallel\Sigma_{i=1}^k\Sigma_{j=1}^l{V_{i,j}}\parallel/b_{m,n}\rightarrow0$$max1≤k≤m,1≤l≤n∥Σi=1kΣj=1lVi,j∥/bm,n→0 a.c. and in ℒp as m ∨ n → ∞ is shown to hold if $$\Sigma_{m=1}^\infty\Sigma_{n=1}^\infty{E}\parallel{V_{m,n}}\parallel^p/{b_{m,n}^p}<\infty$$Σm=1∞Σn=1∞E∥Vm,n∥p/bm,np<∞. This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 < p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space.