Let , be random variables and , , triangular array of constants. Various conditions on and are given, under which almost surely, as
In particular, if the , , are identically distributed, it is sufficient that and for some
This improves the result by Y. S. Chow and T. L. Lai, Ann. Probab. 1, 810-824 (1973; Zbl 0303.60025). In the i.i.d. case, E X and are sufficient.