Long-memory property of nonlinear transformations of break processes

Not much is known about the long-memory property of nonlinearly transformed fractionally integrated processes, let alone of occasional structural break ones. In an important work that fills this gap in the literature, Dittmann and Granger (2002) [Dittmann, I. and Granger, C.W.J., 2002. Properties of nonlinear transformations of fractionally integrated processes. Journal of Econometrics 110, 113-133] show that the estimated long-memory parameter decreases as the probability of break increases, and that it decreases only slightly with the order of the power transformation. In this note, additional results are provided that confirm the latter claim, but contradict the former one.