The so-called "other law of the iterated logarithm" was first given by Chung (Trans. Amer. Math. Soc.64 (1948), 205-233) and Mogul'skii (Theor. Probab. Appl.24 (1979), 405-413) respectively for a sequence of independent random variables and for the standard empirical process. In this work, we...