Abstract: Many probabilists investigated questions on increments of a Wiener process. Hanson and Russo put forward a new kind of increments of a Wiener process, but almost all results are limited to "lim sup". Here, we further investigate it and obtain sereral results on "lim inf". On the basis of these, we farther investigate another kind of results proved by Sheng Zhaoxuan in 5, and obtain some fine results.