Mean convergence theorem for arrays of random elements in martingale type p banach spaces

摘要

For weighted sums of the form S_n = Σ~(v_n)_(j = u_n) a_(nj)(V_(nj) - C_(nj)) where {u_n, n ≥ 1} and {u_n, n ≥ 1} are sequences of integers, {a_(nj), u_n ≤ j ≤ u_n, n ≥ 1} are constants, {V_(nj), u_n j ≤ v_n, n≥1} are random elements in a real separable martingale type p Banach space, and {C_(nj), u_n ≤ j ≤ v_n, n ≥ 1} are suitable conditional expectations, a mean convergence theorem is established. This result takes the forms ||S_n||→~(L_r)0. No conditions are imposed on the joint distributions of the {V_(nj), u_n ≤ j ≤ v_n, n ≥ 1}. The mean convergence theorem is proved assuming that {||V_(nj)||~r, un ≤ j ≤ v_n, n ≥ 1} is {|a_(nj)|~r}-uniformly integrable with respect to {r_n, v_n} which is weaker than Cesaro uniform integrability. The current work extends that of Gut (1992), Adler, Rosalsky and Voldin (1997) and Sung (1999).