A NECESSARY MOMENT CONDITION FOR THE FRACTIONAL FUNCTIONAL CENTRAL LIMIT THEOREM

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. When <italic>d</italic> is close to <inline-graphic>null</inline-graphic> this moment condition is very strong. Our main result is to show that when <inline-graphic>null</inline-graphic> and under some relatively weak conditions on <italic>u</italic>, the existence of <inline-graphic>null</inline-graphic> moments is in fact necessary for the FCLT for fractionally integrated processes and that <inline-graphic>null</inline-graphic> moments are necessary for more general fractional processes. Davidson and de Jong (2000, <italic>Econometric Theory</italic> 16, 643–666) presented a fractional FCLT where only <italic>q</italic> > 2 finite moments are assumed. As a corollary to our main theorem we show that their moment condition is not sufficient and hence that their result is incorrect.