Characterization of Type p Banach Spaces by the Weak Law of Large Numbers

For weighted sums of the form?j=1kn anj Xnj\sum{_{j=1}^{k_(n)}}a_({nj})X_({nj})where{a_(nj),1?j?k_(n)↑∞,n?1}is a real constant array and{X_(aj),1≤j≤k n,n≥1}is a rowwise independent,zero mean,random element array in a real separable Banach space of typep,we establishL r convergence theorem and a general weak law of large numbers respectively,conversely,we characterize Banach spaces of typep in terms of convergence inr-th mean and probability for such weighted sums.