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

AbstractLet {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 0<p<2 and r⩾1∨p, put f(ε)=∑n⩾1nr/p−2P(‖Sn‖⩾εn1/p), ε>0. Jain (1975) [4] proved that f(ε)<∞, ε>0, if and only if E‖X1‖r<∞ and EX1=0. We strengthen this result by showing that, except for the case p=r=1, which is treated separately, ∫δ∞f(ε)dε<∞, δ>0, if and only if E‖X1‖r<∞ and EX1=0