A Necessary Moment Condition for the Fractional Functional Central Limit Theorem

Abstract

We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=Δ^(-d)u(t), where d ∈ (-1/2,1/2) is the fractional integration parameter and u(t) is weakly dependent. The classical condition is existence of q>max(2,(d 1/2)⁻¹) 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 under some relatively weak conditions on u(t), the existence of q≥max(2,(d 1/2)⁻¹) is in fact necessary for the FCLT for fractionally integrated processes and that q>max(2,(d 1/2)⁻¹) moments are necessary and sufficient for more general fractional processes. Davidson and de Jong (2000) presented a fractional FCLT where only q>2 finite moments are assumed, which is remarkable because it is the only FCLT where the moment condition has been weakened relative to the earlier condition. As a corollary to our main theorem we show that their moment condition is not sufficient.