On the weak laws for arrays of random variables

The convergence in probability of the sequence of sums is obtained, where {un,n[greater-or-equal, slanted]1} and {vn,n[greater-or-equal, slanted]1} are sequences of integers, {Xni,un[less-than-or-equals, slant]i[less-than-or-equals, slant]vn,n[greater-or-equal, slanted]1} are random variables, {cni,un[less-than-or-equals, slant]i[less-than-or-equals, slant]vn,n[greater-or-equal, slanted]1} are constants or conditional expectations, and {bn,n[greater-or-equal, slanted]1} are constants satisfying bn-->[infinity] as n-->[infinity]. The work is proved under a Cesàro-type condition which does not assume the existence of moments of Xni. The current work extends that of Gut (1992, Statist. Probab. Lett. 14, 49-52), Hong and Oh (1995, Statist. Probab. Lett. 22, 52-57), Hong and Lee (1996, Bull. Inst. Math. Acad. Sinica 24, 205-209), and Sung (1998, Statist. Probab. Lett. 38, 10-105).