We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of <italic>x</italic> = Δ⁻ <italic>u</italic>, where <inline-graphic>null</inline-graphic> is the fractional integration parameter and <italic>u</italic> is weakly dependent. The classical condition is existence of <italic>q</italic> ≥ 2 and <inline-graphic>null</inline-graphic> moments of the innovation sequence....

This paper presents a family of simple nonparametric unit root tests indexed by one parameter, <italic>d</italic>, and containing the Breitung (2002, <italic>Journal of Econometrics</italic> 108, 342–363) test as the special case <italic>d</italic> = 1. It is shown that (a) each member of the family with <italic>d</italic> 0 is consistent, (b) the asymptotic...