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 xt = Δ−dut, where $d\, \in \,\left({ - {1 \over 2}\,,\,{1 \over 2}} \right)$ is the fractional integration parameter and ut is weakly dependent. The classical condition is existence of q ≥ 2 and $q\, > \,\left( {d\, + \,{1 \over 2}} \right)^{ - 1} $ moments of the innovation sequence. When d is close to $ - {1 \over 2}$ this moment condition is very strong. Our main result is to show that when $d\, \in \,\left({ - \,{1 \over 2},\,0} \right)$ and under some relatively weak conditions on ut, the existence of $q\, \ge \,\left({d\, + \,{1 \over 2}} \right)^{ - 1} $ moments is in fact necessary for the FCLT for fractionally integrated processes and that $q\, > \,\left( {d\, + \,{1 \over 2}} \right)^{ - 1} $ moments are necessary for more general fractional processes. Davidson and de Jong (2000, Econometric Theory 16, 643–666) 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.