Let B be a p type Banach space (1<p<2),φ(x) be some slow increasing functions on [0,+∞), if {X, X n,n≥1} is a sequence of i.i.d B valued random variables, then we obtain a necessary and sufficient condition for the moments of Sup n ‖S n(nφ(n)) 1/p ‖ being bounded.