On the weak laws for arrays of random variables

The convergence in probability of the sequence of sums &USigma;(vn)(i=un) (X-ni - c(ni)/b(n) is obtained, where {u(n,) n ≥ 1} and {v(n) n ≥ 1} are sequences of integers, {X-ni, u(n) ≤ i ≤ v(n), n ≥ 1) are random variables, {c(ni), u(n) ≤ i ≤ v(n), n ≥ 1} are constants or conditional expectations, and (b(n), n ≥ 1) are constants satisfying b(n) → ∞ as n → ∞. The work is proved under a Cesdro-type condition which does not assume the existence of moments of X,,i. 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).) © 2005 Elsevier B.V. All rights reserved.