Improved convergence rates for tail probabilities for sums of i.i.d. Banach space valued random vectors

Let { X n , n ⩾ 1 } be a sequence of i.i.d. random vectors taking values in a 2-smooth separable Banach space, and set S n = X 1 + ⋯ + X n . For 0 < p < 2 and r ⩾ 1 ∨ p , put f ( ε ) = ∑ n ⩾ 1 n r / p − 2 P ( ‖ S n ‖ ⩾ ε n 1 / p ) , ε > 0 . Jain (1975) [4] proved that f ( ε ) < ∞ , ε > 0 , if and only if E ‖ X 1 ‖ r < ∞ and E X 1 = 0 . We strengthen this result by showing that, except for the case p = r = 1 , which is treated separately, ∫ δ ∞ f ( ε ) d ε < ∞ , δ > 0 , if and only if E ‖ X 1 ‖ r < ∞ and E X 1 = 0 .