Strong Laws of Large Numbers for Double Sums of Banach Space Valued Random Elements

For a double array {V_m,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 {b_m,n, m≥1,n ≥ 1} of positive constants, the limit law max_{1≤k≤m,1≤l≤n}||Σ _i=1^k||Σ _j=1^l V_i,j||/b_m,n → 0 a.c. and in L_p as m ∨ n → ∞ is shown to hold if Σ_m=1^∞ Σ_n=1^∞ E||V_m,n||^p/b_m,n^p < ∞. 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.