On the complete convergence of sequences of random elements in Banach spaces.

For a sequence { X n , n ⩾ 1 } of independent random elements taking values in a Rademacher type p Banach space with the k-th partial sum S k (k ⩾ 1) , we provide necessary and sufficient conditions for the convergence of ∑ n = 1 ∞ 1 n P (max 1 ⩽ k ⩽ n ‖ S k ‖ > ε n α) and ∑ n = 1 ∞ log n n P (max 1 ⩽ k ⩽ n ‖ S k ‖ > ε n α) for every ε > 0 . [ABSTRACT FROM AUTHOR]