In this paper, we study the asymptotics of nonstationary fractional integrated time series, the long memory time series with d ≥ ½, with special attention focused on the cases when d = (2p + 1)/2 for integer n no less than 0. There is considerable empirical evidence showing long memory of this magnitude in many economic time series including the inflation rate and the stock market volatility. A study of the large-sample property is therefore both needed and useful. Also, we found the asymptotics of nonstationary fractional integrated time series useful in the study of the large-sample theory of the Kwiatkowski–Phillips–Schmidt–Shin test (1992, Journal of Econometrics 54, 159–178).