Dependence of a class of non-integer power functions

This short article exhibits that there exists critical point of the power for the generalized function t-a for a > 0. The present results show that it is long-range dependent if 0 < a < 1 and short-range dependent when a > 1. My motivation of studying that dependence issue comes from the power-law type functions in fractal time series. The present results may yet be useful to investigate fractal behavior of fractal time series from a new point of view.