A Necessary Moment Condition for the Fractional Central Limit Theorem

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We discuss the moment condition for the fractional functional central limit theorem (FCLT) for partial sums of x(t)=¿^{-d}u(t) , where -1/2<d<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 -1/2<d<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, Econometric Theory 16, 643-- 666) presented a fractional FCLT where onlyq>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.
TidsskriftEconometric Theory
Udgave nummer28
Sider (fra-til)671-679
Antal sider9
StatusUdgivet - 2012

ID: 38043678