Let {Xni:1≤i≤n,n≥1} be an array of rowwise independent B-valued random variables, and let g(x) be a regular varying function with index 1/P(P>0) . Let {ani:1≤i≤n,n≥1} be an array of real numbers. Let r>0 . The sufficient conditions such that (?) are obtained. The wellknown results by Stout and Sung etc. are extended.