Let B be a p-type Banach space (1<p <2) ,φ( x ) be some slow increasing functions on [0,+∞) ,if {X ,
Xn,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 Supn ||Sn/(nφ(n))1/p || being bounded.