A necessary moment condition for the fractional functional central limit theorem - Søren Johansen; Morten Ørregaard Nielsen (Queen's University, Department of Economics, 2010)
We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x_t=Delta^-du_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>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.
Keywords: C22; 330; fractional integration; functional central limit theorem; long memory; moment condition; necessary condition; Zeitreihenanalyse; Statistische Methode; Theorie