In this paper, we derive the asymptotic distribution of the Augmented Dickey-Fuller t Test statistics, t_{ADF}, against a generalized fractional integrated process (for example: ARFIMA(p,1+d,q) ,|d|<1/2,and p, q be positive integer) by using the propositions of Lee and Shie (2003). Then we discuss why the power decreases with the increasing lags in the same and large enough sample size T when d is unequal to 0. We also get that the estimator of the disturbance's variance, S^2, has slightly increasing bias with increasing k. Finally, we support the conclusion by the Monte Carlo experiments.