In this paper, we study the asymptotics of nonstationary fractionalintegrated time series, the long memory time series withd ≥ ½, with special attention focused on thecases when d = (2p + 1)/2 forinteger n no less than 0. There is considerableempirical evidence showing long memory of this magnitude in manyeconomic time series including the inflation rate and the stockmarket volatility. A study of the large-sample property is thereforeboth needed and useful. Also, we found the asymptotics ofnonstationary fractional integrated time series useful in the studyof the large-sample theory of the Kwiatkowski–Phillips–Schmidt–Shintest (1992, Journal of Econometrics 54,159–178).