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

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 ■ and in Lp as m∨n→∞ is shown to hold if ■ 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.