Robert PARKER, Andrew ROSALSKY
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 max1≤k≤m,1≤l≤n||Σ i=1k||Σ j=1l Vi,j||/bm,n → 0 a.c. and in Lp as m ∨ n → ∞ is shown to hold if 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.
