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

Let {Xn,n⩾1} be a sequence of i.i.d. random vectors taking values in a 2-smooth separable Banach space, and set Sn=X1+⋯+Xn. For 00. Jain (1975) [4] proved that f(ε)<∞, ε>0, if and only if E‖X1r‖<∞ and EX1=0. We strengthen this result by showing that, except for the case p=r=1, which is treated separately, , δ>0, if and only if E‖X1r‖<∞ and EX1=0.