A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces

For weighted sums of the form S-n=Sigma(j=l)(kn)a(nj)(V-nj-c(nj)) where {a(nj),1 less than or equal to j less than or equal to k(n) < infinity, n greater than or equal to 1} are constants, {V-nj, 1 less than or equal to j<less than or equal to n greater than or equal to 1} are random elements in a real separable martingale type p Banach space, and {c(nj), 1 less than or equal to j less than or equal to k(n), n greater than or equal to 1} are suitable conditional expectations, a mean convergence theorem and a general weak law of large numbers are established. These results take the form \\S-n\\ -->(Lr) 0 and S-n -->(P) 0, respectively. No conditions are imposed on the joint distributions of the {V-nj, 1 less than or equal to j less than or equal to k(n), n greater than or equal to 1}. The mean convergence theorem is proved assuming that {\\V-nj\\(r), 1 less than or equal to j less than or equal to k(n), n greater than or equal to 1} is {\a(nj)\(r)}-uniformly integrable whereas the weak law is proved under a Cesaro type condition which is weaker than Cesaro uniform integrability. The sharpness of the results is illustrated by an example. The current work extends that of Gut (1992) and Hong and Oh (1995).