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 x_{t} = Delta^{-d} u_{t}, where d in (-1/2,1/2) is the fractional integration parameter and u_{t} is weakly dependent. The classical condition is existence of q≥2 and q>1/(d+1/2) moments of the innovation sequence. When d is close to -1/2 this moment condition is very strong. Our main result is to show that when d in (-1/2,0) and under some relatively weak conditions on u_{t}, the existence of q≥1/(d+1/2) moments is in fact necessary for the FCLT for fractionally integrated processes, and that q>1/(d+1/2) moments are necessary for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>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.