A convergence theorem for sums of dependent Hilbert space valued triangular arrays

A generalization of Chung's (1974) theorem concerning a.s. convergence for sums of triangular arrays is proved under conditions weaker than zero correlation. A theorem of Ahn (1988) and a theorem of Jobson and Fuller (1980) can then be proved. Originally these two theorems were proved by a lemma of Jobson and Fuller (1980), but there exists a counter example for this lemma.